Topics on Perfect Graphs (Annals of Discrete Mathematics, Volume 21)

TOPICS ON PERFECT GRAPHS annals of discrete mathematics General Editor Peter L. HAMMER, Rutgers University, New Brun...

Author: Claude Berge | Vasek Chvatal

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TOPICS ON PERFECT GRAPHS

annals of discrete mathematics General Editor

Peter L. HAMMER, Rutgers University, New Brunswick, NJ, U.S.A. Advisory Editors

C. BERGE, UniversitC de Paris, France M.A. HARRISON, University of California, Berkeley, CA, U.S.A. V. KLEE, University of Washington, Seattle, WA, U.S.A. J.H. VAN LINT, California Institute of Technology, Pasadena, CA, U.S.A. G.-C. ROTA, Massacbusetts Institute of Technology, Cambridge, MA, U.S.A.

NORTH-HOLLAND - AMSTERDAM

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NEW YORK

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NORTH-HOLLAND MATHEMATICS STUDIES

88

Annals of Discrete Mathematics (21) General Editor: Peter L. Hammer Rutgers University, New Brunswick, U.S.A.

Topics on Perfect Graphs Edited by

C. BERGE E. R. Combinatoire, Centre de Mathhatique Sociale, Paris, France

v.

CHVATAL

Department of Computer Science, McGill University, Montreal, Canada

1984 NORTH-HOLLAND - AMSTERDAM

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@ Elsevier Science Publishers B.V. 1984

All rights reserved. No part of this publication may be reproduced, stored in a retrieval sysiem or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 86587 X

Pitblisliers: ELSEVIER SCIENCE PUBLISHERS B.V. P. 0 . BOX 1991 1000 BZ AMSTERDAM T H E NETHERLANDS Sole distributors for the U . S .A . and Carinda: ELSEVIER SCIENCE PUBLISHING COMPANY, INC 52 VANDERBILTAVENUE NEWY0RK.N.Y. 10017 U.S.A.

Library of Cougres Cataloging in Publication Data Main entry under title: Topics on perfect graphs. (North-Holland mathematics studies; 88) (Annals of discrete mathematics; 21) 1. Perfect graphs. 1. Berge, Claude. 11. Chvatal, V. (Valclav) 111. Series. IV. Series: Annals of discrete mathematics; 21. QA166.16.T66 1983 51 1 ’ 5 8-3-2353 ISBN 0-444-86587-X (US.)

PRINTED IN THE NETHERLANDS

CONTENTS Introduction

vii

PART I. General results

1

C. BERGE,Minimax theorems for normal hypergraphs and balanced hypergraphs - A survey

3

J.-C. FOURNIER and M. LAS VERGNAS, A class of bichromatic hypergraphs

21

L. LOVASZ, Normal hypergraphs and the Weak Perfect Graph Conjecture

29

PART 11. Special classes of perfect graphs

43

C. BERGE,Diperfect graphs

45

C. BERGEand P. DUCHET,Strongly perfect graphs

57

V. CHVATAL, Perfectly ordered graphs

63

P. DUCHET,Classical perfect graphs

67

C. GRINSTEAD, The Perfect Graph Conjecture for toroidal graphs

97

W.-L. Hsu, The Perfect Graph Conjecture on special graphs - A survey

103

H. MEYNIEL, The graphs whose odd cycles have at least two chords

115

E. OLARUand H. SACHS,Contributions to a characterization of the structure of perfect graphs

121

G. RAVINDRA, Meyniel’s graphs are strongly perfect

145

A. TUCKER,The validity of the Perfect Graph Conjecture for &-free graphs

149

PART 111. Polyhedral point of view

159

R. GILES,L.E. TROTTER, Jr. and A. TUCKER, The Strong Perfect Graph 161 Theorem for a class of partitionable graphs M.W. PADBERG, A characterization of perfect matrices

169

vi

Conrents

PART IV. Which graphs are imperfect

179

R.G. BLAND,H.-C. HUANGand L.E. TROITER, JR. Graphical properties 181 related to minimal imperfection V. CHVATAL,An equivalent version of the Strong Perfect Graph 193 Conjecture

V. CHVATAL, R.L. GRAHAM,A.F. PEROLDand S.H. WHITESIDES, 197 Combinatorial designs related to the Perfect Graph Conjecture S.H. WHITESIDES,A classification of certain graphs with minimal 207 imperfection properties PART V. Which graphs are perfect

219

R.E. BIXBY,A composition for perfect graphs

22 1

M. BURLETand J. FONLUFT,Polynomial algorithm to recognize a Meyniel graph

225

M. BURLETand J.-P. UHRY,Parity graphs

253

V. CHVATAL, A semi-strong Perfect Graph Conjecture

279

S.H. WHITESIDES,A method for solving certain graph recognition and 28 1 optimization problems, with applications to perfect graphs PART VI. Optimizaiion in perfect graphs

299

M.C. GOLUMBIC, Algorithmic aspects of perfect graphs

301

M. GROTSCHEL, L. LovAsz and A. SCHWVER, Polynomial algorithms for perfect graphs

325

W.-L. Hsu and G.L. NEMHAUSER, Algorithms for maximum weight cliques, minimum weighted clique covers and minimum colorings of claw-free perfect graphs 357

INTRODUCTION Many challenging problems in graph theory involve at least one of the following four invariants: (i) the stability number a ( G ) (also called the independence number), defined as the largest number of pairwise nonadjacent vertices in G ; (ii) the clique covering number 8(G), defined as the least number of cliques which cover all the vertices of G ; (iii) the clique number w ( G ) , defined as the largest number of pairwise adjacent vertices in G ; (iv) the chromatic number y ( G ) (sometimes denoted also by x ( G ) ) ,defined as the least number of colors needed to color all the vertices in such a way that no two adjacent vertices have the same color. The inequality a ( G )S O(G)holds trivially for all graphs G : if k cliques cover all the vertices then no more than k vertices can be pairwise nonadjacent. (A similar observation shows that w ( G ) Sy ( G ) .In fact, if G denotes the complement of G then o ( G )= a (G) and y ( G )= 8(C?).) Graphs which satisfy this inequality with the equality sign played an important role in Claude Shannon’s 1956 paper concerning the ‘zero error capacity of a noisy channel’. In this paper, Shannon remarked that the smallest graph G with a ( G ) < 8(G) is G, the cycle of length five. It was Shannon’s work which motivated Claude Berge to make a conjecture (first presented at a graph theory meeting organized by Horst Sachs in Halle an der Saale in March 1960) concerning graphs with a ( G )= 8 ( G ) .This conjecture may be stated in many different ways which are easily seen to be equivalent. Three of them go as follows. (Sl) If a graph G has no induced subgraph isomorphic to either the chordless cycle C, whose length p is odd and at least five or the complement of such a cycle, then a ( G )= 8 ( G ) .

c,

(S2) The only minimal graphs G with a ( G )< 8 ( G )are the Cp’sand the cp’s with p odd and at least five. (S3) A graph G satisfies a ( G A )= f3(GA)for every set A of vertices (with GA standing for the subgraph of G induced by A ) if and only if no GA is isomorphic to a C, or a with p odd and at least five.

c,

An early effort by Alain Ghouila-Houri failed to produce a counterexample to this conjecture. Despite this encouraging sign, Berge felt that the conjecture might be too ambitious. Therefore he restricted himself to a weaker conjecture in the hope that it might be easier to settle. Again, this conjecture may be stated in many different ways which are easily seen to be equivalent; we list only three. vii

viii

Introduction

(W1) If a graph G satisfies a ( G A )= B(GA)for every set A of vertices then it satisfies y ( G ) = w(G). (W2) If a graph G satisfies y(GA) = w ( G A )for every set A of vertices then it satisfies a ( G ) = B(G). (W3) The class of graphs satisfying a(GA) = O(GA)for all A is closed under complementation. Clearly, the conjecture (W) is weaker than the conjecture (S). For this reason,

(S)and (W) became known as The Strong Perfect Graph Conjecture and The Weak Perfect Graph Conjecture, respectively. (Anticipating in this brief historical sketch, we note that the Weak Perfect Graph Conjecture was proved by Lovslsz in 1971. Nowadays, it is known as the Perfect Graph Theorem.) Yet another way of phrasing (W) is to say that ‘a graph is a-perfect if and only if it is y-perfect’, with ‘a-perfect’ and ‘y-perfect’ defined as ‘satisfying the hypotheses of (Wl) and (W2)’, respectively. (LovBsz’s proof of (W) made this terminology obsolete: since ‘a-perfect’ and ‘y-perfect’ are synonymous, both of them may be replaced by ‘perfect’.) The evolution of the theory of perfect graphs may be traced back to the first international meeting on graph theory held at Dobogoko (Hungary) in October 1959. At this meeting, A. Hajnal and J. SurBnyi presented an elegant result: Every triangulated graph G satisfies a ( G )= B(G).(An immediate corollary of this theorem states that every triangulated graph G is a-perfect.) Berge complemented this result by showing that every triangulated graph G satisfies y(G) = w(G). (Now the easy corollary states that every triangulated graph is y-perfect.) Berge also noticed that there are other interesting classes of graphs which are simultaneously a-perfect and y-perfect. For instance, the comparability graphs are a-perfect by virtue of Dilworth’s theorem and y-perfect by an easy ad hoc argument. Similarly, the line graphs of bipartite graphs are a-perfect and y-perfect by two different theorems of Konig. After the meeting at Halle an der Saale in 1960, the Strong Perfect Graph Conjecture received the enthusiastic support of G. HBjos and T. Gallai. In fact, Gallai provided further evidence in support of the conjecture by strengthening the results on triangulated graphs: he proved that a graph is a-perfect and y-perfect whenever each of its odd cycles of length at least five has at least two non-crossing chords. Nevertheless, Berge still felt that the weak conjecture was more promising. At a conference at Rand Corporation in the summer of 1961, he had fruitful discussions with Alan Hoffman, Ray Fulkerson and others. Later on, discussions between Alan Hoffman and Paul Gilmore led Gilmore to a rediscovery of the Strong Perfect Graph Conjecture and to an attempt to axiomatize the relevant

Introduction

ix

properties of cliques in perfect graphs. Ray Fulkerson attacked the weak conjecture from a linear programming point of view, which led to the development of his theory of ‘antiblocking polyhedra’. He proved that the conjecture was equivalent to another statement, which he found too strong to be true. For this reason, he concentrated his efforts on attempts to find a counterexample: even though he had reduced the conjecture to a certain ‘duplication lemma’, he missed its proof. (Later, when informed by Berge that the validity of the conjecture had just been established by Loviisz, he was able to supply the missing link independently in only a few hours.) Lovhsz’s beautiful proof of the Weak Perfect Graph Conjecture was found in 1971 independently of Fulkerson’s work. After more than twenty years, the Strong Perfect Graph Conjecture remains open. The question of its validity alone (or the problem of describing all minimal imperfect graphs) has become only secondary when compared with the important body of work stimulated by the conjecture over the years. Much of this work has an intrinsic interest independent of the Strong Perfect Graph Conjecture: it would not become obsolete even if the conjecture were proved. The purpose of this book is to present selected results on perfect graphs in a single volume. These are reprinted classical papers (sometimes with slight simplifications), survey papers written for this collection or new results. They concern different, and often overlapping, aspects of perfect graphs. We shall now comment on some of these aspects.

Part I. General Results When described by a reference to its cliques, a perfect graph becomes a normal hypergraph. LovBz’s proof of the Perfect Graph Theorem (1972) is given on pp. 29-42 of this volume in its original form, first in the context of hypergraph theory, and then with another characterization. A proof of Lovhsz’s theorem also appears in the article on pp. 3-19, together with various minimax equalities for normal hypergraphs - and, more specifically, for balanced hypergraphs, which motivated, since 1970,. the development of hypergraph theory. The elegant framework of hypergraph theory is also the setting of a work by Fournier and Las Vergnas (pp. 21-27), who proved in 1972 an interesting property of cliques in perfect graphs conjectured by Lovhsz.

Part II. Special Classes of Perfect Graphs Three classical examples (triangulated graphs, comparability graphs and line graphs of bipartite graphs) have been mentioned above. Results in this direction are surveyed in the papers by Duchet (pp. 67-96) and Golumbic (pp. 301-323).

Introduction

X

Another classical example is provided by the theorem of Gallai: A graph is perfect whenever each of its odd cycles of length at least five has two non-crossing chords. A companion theorem, in which ‘non-crossing’is replaced by ‘crossing’, was proved by Olaru and may be found in the paper by Olaru and Sachs (pp. 121-144). Both of these results were generalized by Meyniel in 1976 (see article this volume, pp. 115-119): A graph is perfect if each of its odd cycles of length at least five has two (or more than two) chords. Other special classes of perfect graphs may be obtained by forbidding, in addition to all the Cp’sand cp’s with p odd and at least five, an extra graph F. This has been done for F = K4 by Tucker (see pp. 149-157), and for F = K1,3by Parthasarathy and Ravindra (see articles on pp. 103-113 and pp. 161-167). The paper by Hsu (see pp. 103-113) generalizes various techniques used in this direction. As examples of ‘ad hod classes, we have included the paper of Grinstead (pp. 97-101), concerning toroidal graphs. In the first article in this volume, the reader will also find the main results about another class, the balanced graphs, whose perfectness follows from a theorem of Berge and Las Vergnas (1970): A graph is perfect whenever each odd cycle has an edge inducing only maximal cliques which contain three vertices of the cycle. Finally, a new class, the ‘strongly perfect’ graphs, is introduced by Berge and Duchet (pp. 57-61): In every induced subgraph of a strongly perfect graph, some stable set meets all the maximal cliques. This class includes the comparability graphs, the triangulated graphs, and the complements of triangulated graphs. Ravindra (pp. 145-148) and Chv6tal (pp. 63-65) have shown that it includes also the Meyniel graphs and the ‘perfectly orderable’ graphs, respectively.

Part III. Polyhedral Point of View With each graph G on n vertices, we may associate two polytopes P ( G ) and Q ( G ) .The first polytope P ( G ) is the convex hull of all the incidence vectors of stable sets in G (a stable set being a set of pairwise nonadjacent vertices). The second polytope is obtained by associating a variable x, with each vertex of G and then defining O(G)by the system of inequalities xu s 1 for each clique C, V E C

x, 3 0 for each vertex u.

Trivially, P ( G ) C Q ( G ) for every graph G. The unique character of perfect graphs is illuminated by the fact that P ( G )= Q ( G )if and only if G is perfect. This statement has been proved first by Fulkerson and then independently by Chvital; yet another proof may be found in Berge (pp. 3-19). Further results in

Introduction

xi

this direction have been found by Padberg (see pp. 169-178). Padberg’s results are used by Giles, Trotter and Tucker (pp. 161-167) to establish a class of graphs for which the Strong Perfect Graph Conjecture holds true.

Part IV. Which Graphs are Imperfect? If the Strong Perfect Graph Conjecture holds true, then the answer is ‘those containing a C, or a with p odd and at least five’; in any case, the question is equivalent to asking which graphs are minimal imperfect. A major breakthrough in this direction is due to Lovhsz (see article on pp. 29-42): Every minimal imperfect graph G with n uertices has n = (Y ( G ) w ( G )+ 1. This result led Bland, Huang and Trotter (pp. 181-192) to call a graph G partitionable if there are integers r, s greater than one and such that (i) G has precisely rs + 1 vertices; (ii) for each vertex u of G, the vertex-set of G - u can be partitioned into r disjoint cliques of size s and into s disjoint stable sets of size r. Bland, Huang and Trotter observed that a graph is imperfect if and only if it contains an induced partitionable subgraph. (The ‘only if’ part follows instantly from Lovisz’s theorem; to see the ‘if’ part, note that (ii) along with r,s 2 2 implies ( Y ( G ) =r, w ( G ) = s.) Jack Edmonds and Kathie Cameron (K. B, Cameron, Polyhedral and Algorithmic Ramifications of Antichains, Ph.D. Thesis, University of Waterloo, 1982) pointed out an immediate corollary of this observation: the class of imperfect graphs belongs to NP. Building up on Lovhsz’s theorem, Padberg (pp. 169-178) was able to establish additional properties of minimal imperfect graphs. Each of them (with n vertices) has precisely n stable sets S1,S2,.. . ,S, of size (Y = a ( G )and precisely n cliques C1,C,, . . . ,C, of size w = w ( G ) . Furthermore, each vertex is in precisely (Y stable sets S, and in precisely w cliques C,. Finally, S, n C, = 0 if and only if i = j (for some appropriate choice of indexing). For every choice of a and w greater than one, a graph satisfying Padberg’s conditions may be constructed by taking vertices u l , uz, . . . ,v, ( n = a w + 1) and making u, adjacent to u, if and only if 1 i - j 1 < w (with arithmetic modulo n ) . The resulting graph is denoted by CZil. A theorem found by ChvGtal (see pp. 193-195) shows that the Perfect Graph Conjecture may be restated as follows:

c,

(S4) Every minimal imperfect graph G has a spanning subgraph isomorphic to C:::l with (Y = (Y ( G ) and w = w ( G ) . Unfortunately, Padberg’s conditions may be satisfied by graphs radically different from CZ:l; Chvhtal, Graham, Perold and Whitesides (see pp. 197-206) described ways of constructing infinite families of graphs which, in spite of their

xii

Introduction

unwieldy structure, do satisfy Padberg’s conditions. Two of these graphs were found independently by Bland, Huang and Trotter (see pp. 181-192). The case of a = 4 and w = 3 is studied in detail by Whitesides (see pp. 207-218).

Part V. Which Graphs are Perfect? From the point of view of computational complexity, this question is definitely not just another way of asking which graphs are imperfect. T o this day, nobody has even guessed at a ‘certificate of perfection’ which could be attached to every perfect graph and whose validity could be checked in polynomial time. We believe that even just a correct guess at such a certificate (a companion to the Strong Perfect Graph Conjecture) would bring us a long way towards settling the Strong Perfect Graph Conjecture itself. Still, the question as to whether such a certificate exists at all is rarely asked. Analogous questions are answered satisfactorily for many special classes of perfect graphs; the details may be found in Golumbic (pp. 301-323). For instance, there is a polynomial-time algorithm which, given an arbitrary graph G, will find out whether G is a comparability graph or not. In fact, this algorithm will also furnish a certificate for its output: either a certain way of putting arrows on the edges of G, which certifies that G is a comparability graph, or a certain sequence of vertices in G, which certifies that G is not a comparability graph. A companion polynomial-time algorithm will accept any comparability graph along with its certificate (directions on edges), producing a largest clique and a minimum coloring as the output. It is tempting to speculate that the same pattern could be followed by a proof of the Strong Perfect Graph Conjecture. A polynomial-time algorithm, given any graph G, would produce either a certificate of perfection for G or one of the forbidden induced subgraphs (C, or with p odd and at least five in case the Strong Perfect Graph Conjecture is valid). Then a companion polynomial-time algorithm, given any perfect graph along with its certificate of perfection, would produce a largest clique and a minimum coloring. (Of course. the existence of the first of these two algorithms would imply that the class of perfect graphs belongs to P rather than merely to NP.) It is conceivable that every perfect graph can be built from ‘primitive’ perfect graphs by simple operations which preserve perfection. For instance, the role of the primitive perfect graphs could be played by comparability graphs or line graphs of bipartite graphs; the operations could consist of pasting two perfect graphs together along a clique or taking their ‘join’, as in the paper by Bixby (pp. 221-224). If the initial class of ‘primitive’ perfect graphs belonged to NP and if the operations could be carried out in a polynomial time, then the desired ‘certificate of perfection’ would follow immediately. This idea was put forth by Sue Whitesides in conversations with Chvgtal in the fall of 1977 (although it is so

c,

Introduction

...

Xlll

natural that we would not be surprised if others had proposed it earlier). in fact, it motivated her to design the algorithm for recognizing graphs with cliquecutsets, which is reproduced here (pp. 281-297). A systematic and steady progress towards answering the question ‘which graphs are perfect?’ is being made by the Grenoble School. First, Michel Burlet and Jean-Pierre Uhry (pp. 253-277) designed a polynomial-time algorithm to recognize the graphs whose perfection was established by Olaru (every odd cycle of length at least five has two crossing chords), then Burlet, and independently Whitesides, found an analogous result for the graphs of Gallai (every odd cycle of length at least five has two non-crossing chords) and quite recently Burlet and Fonlupt (pp. 225-252) designed a polynomial algorithm to recognize the Meyniel graphs (every odd cycle of length at least five has two chords). This last result implies that every Meyniel graph can be built from ‘primitive Meyniel graphs’ by an interesting new operation which preserves perfection. An intriguing sidelight to the question ‘which graphs are perfect?’ is provided by speculations about its possible interplay with the Perfect Graph Theorem: a certificate of perfection for G provides a certificate of perfection for the complement This point of view led Chvatal (pp. 279-290) to propose a ‘Semi-strong Perfect Graph Conjecture’ which suggests that a certificate of perfection could be formulated in terms of induced P4’s,and therefore apply automatically to a graph and its complement at the same time.

c.

Part VI. Optimization in Perfect Graphs The four invariants cu(G), 8(G), y(G) and w ( G ) are difficult to evaluate: each of the four problems of recognizing graphs G and integers k with a ( G ) ak, B(G)S k, w ( G ) > k and y ( G ) S k, respectively, is NP-complete. (In fact, the second and the fourth problems remain NP-complete even if the value of k is fixed, as long as it is at least three.) Nevertheless, polynomial-time algorithms for solving the four optimization problems (finding a largest set of pairwise nonadjacent vertices, etc.) in restricted classes of perfect graphs have been known for a long time. The classical ones, involving triangulated graphs and comparability graphs, are surveyed in Golumbic (pp. 301-323). Recent results of Burlet and Fonlupt, applying to the wide class of Meyniel graphs, are as yet unpublished and will appear elsewhere in a companion paper to that on pp. 225-252; nevertheless, the algorithms of Burlet and Uhry (see pp. 253-277) for solving the four optimization problems in the class of Olaru-Sachs graphs may be found in this volume. Hsu and Nemhauser (see pp. 357-369) treat the class of claw-free perfect graphs. A fundamental result in this direction is due to Grotschel, Lovasz and Schrijver (see pp. 325-356), who designed polynomial-time algorithms for

xiv

Introduction

solving the four optimization problems in arbitrary perfect graphs. Their algorithms are ingenious variations on the celebrated ‘ellipsoid method’ for linear programming, and therefore very much unlike the typical combinatorial optimization procedures. Can they be replaced by polynomial-time algorithms of a more transparent combinatorial nature?

PART I

GENERAL RESULTS

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Annals of Discrete Mathematics 21 (1984) 3-19 @ Elsevier Science Publishers B.V.

MINIMAX THEOREMS FOR NORMAL HYPERGRAPHS AND BALANCED HYPERGRAPHS - A SURVEY C . BERGE C.N. R.S. Paris

1. Introduction

By definition, a minimax property can be written as follows: “the minimum of something is equal to the maximum of something else”. For the perfect graph G, the properties “ y ( G )= u(G)” and “cu(G) = 6(G)”are of that type. In fact, the perfect graphs have several other minimax properties which are usually stated in the terminology of “normal hypergraphs”. In 1969, we introduced a special class of perfect graphs defined as follows: for every odd cycle (el, e2,. . .,elk+,), and every sequence of distinct maximal cliques (C,,C,,. . . , Czk+I), where Cicontains the two end-points of e i , at least one of the C,’s contains three vertices of the cycle. This concept, also called “balanced hypergraph”, has been of some importance for the theory of linear programming in integers and generalizes the totally unimodular (0,1)-matrices. This paper is intended to survey and to complete the collection of minimax properties for normal hypergraphs and balanced hypergraphs. We shall remain in the context of Hypergraph Theory, but the reader can easily translate all the results in terms of graphs. In Section 2, we recall the definitions of Hypergraph Theory which are related to packing problems. In Section 3, we give the main minimax equalities for normal hypergraphs in only one theorem; a unified proof is given for Lovkz’s theorem [19], Chviital’s theorem [5],and other results. This section is mainly based on results of LOV~SZ, which give a more elegant presentation than the theory of antiblocking polyhedra (see [13]). In Section 4, we study the hypergraphs with the Menger property. These hypergraphs are not normal but have similar properties. In Section 5, we study the paranormal hypergraphs. In Section 6, we prove several minimax theorems concerning balanced hypergraphs. In particular we give an answer to a problem raised by Fulkerson, Hoffman and Oppenheim [14].

3

C. Berge

4

2. General definitions

A hypergraph H is a family (El,E2... . , E m )of non-empty subsets, called edges; U E , = X is the vertex-set, and H is often described by its incidence matrix, i.e., a (0, 1)-matrix A with m columns representing the edges and n rows representing the vertices. This matrix A has no 0-vector as a row or as a column. The rank of H is r ( H ) = max, I E, 1, and the anti-rank is s ( H )= min, I E, 1. The maximum degree A ( H ) is the maximum number of edges having a point in common. A partial hypergraph of H is a hypergraph H’ obtained from H by removing some of the edges (and the vertices which become isolated), or, equivalently, by removing some columns of the incidence matrix A, and the rows which become 0-vectors. The subhypergraph of H induced by a set S C X is the S, land by removing hypergraph Hs obtained by replacing each edge E, by E, f an edge E, if E, n S = 0. The dual hypergraph of H is the hypergraph H * defined by the transpose A * of the incidence matrix A. A set T C X is a transversal set of H if T meets all the edges; the family of all the minimal transversal sets is called the transversal hypergraph, and is denoted by TrH. min{ I TI T E TrH} is called the transuersal number, and is denoted by r ( H ) . A matching is a partial hypergraph H’ of maximum degree 1. The matching number v ( H ) is the maximum number of edges in a matching. If v ( H )= T ( N ) , the hypergraph H is said to have the Kiinigproperty. A transversal T can also be defined by its characteristic vector t = ( t l , tz, . . .,t ” ) , where t, = 1 if T 3 x,, and t, = O otherwise; such a vector t is a (0,l)-vector of the polytope

I

P

={t

I t E R ” , t 2 0, tA

1).

Similarly, a matching H‘ can be defined by a characteristic vector z = (z,, zl,.. ., z m ) ; such a vector z is a (0, 1)-vector of the polytope

O={z IzER”. 220, ArS1). Hence, P is called the transversal polytope and 0 is called the matchingpolytope. If p = ( p , ,p 2 , . . . ,p.) is a vector with non-negative integral coordinates, we define a p-matching as an integral vector of the polytope

0,= { z l z E R “ ,

z S O , Az S p f .

If 9 = (4,. 92,. . .,9,,,) is a vector with non-negative integral coordinates, we define a q-transversal as an integral vector of the polytope P,={tItER”, (30, tAsq}.

The maximum 9-value of a p-matching is denoted by v ( ~p.;4 ) = max{(q, z )

I z E N ” n 0,).

5

Minimax theorems

The minimum p-value of a q-transversal is denoted by

I

T ( H ;p , q ) = min{(p, t ) t E N " fl P,}. Clearly, T ( H ;1,1) = T ( H )and v(H; 1,l)= v(H). Proposition 2.1. We have

w ;P? 4 ) s

7 * ( H P, ;

4 )s

7w;P, q 1,

I

I

where 7 * ( H ; p , q )= max{(q, z ) z E Q,} = min{(p, t ) t E P,}. This follows immediately from the duality principle of linear programming.

3. Normal hypergraphs Let H be a hypergraph. Denote by & ( x ) the degree of a vertex x, put A ( H )= max dH( x ) , and denote by q ( H ) the chromatic index, that is, the least number of colors needed to color the edges so that no two intersecting edges have the same color. Clearly, q ( H )2 A ( H ) . H is called a normal hypergraph if every partial hypergraph H' of H satisfies q ( H ' )= A (H'). It is not difficult to see that H is a normal hypergraph if and only if its dual is the clique-hypergraph of a perfect graph. To study the properties of normal hypergraphs, the basic result is the following (LovBsz's) lemma: Lemma (Lovasz [19]). Let H = ( E 1 , E 2..., , E m ) be a hypergraph on X = { x l ,x 2 , . . . ,x " } , and let y = ( y l , y 2 , .. . ,y m )E N". Then the hypergraph obtained from H by multiplying each E, by y i is also normal.

Proof. Consider the hypergraph fi = (EI,E l , E2,E,, . . . , E m )where E ; = E l ; it suffices to show that q ( f i )= A (fi). Put q ( H )= A ( H ) = q. Case 1. The edge E l contains a vertex x with dH( x ) = A ( H ) .Then, A (I?) = q + 1, and

A (fi)< q (B)< q ( H )+ 1 = q

+ 1 = A (R).

So q ( f i ) = A ( f i ) and the proof is achieved. Case 2. The edge El contains no vertex x with d H ( x )= A ( H ) . Consider an optimal q-coloring of the edges of H. Let (1) be the color received by the edge El. Let H I be the family of edges of H having color (1) which are different from El.Each vertex x with d H ( x )= A ( H )belongs to an edge of HI,so A ( H - Hi) = 4 - 1. Since H is normal, q ( H - HI)= q - 1; therefore, we can color with q - 1

C. Beige

6

colors the edges in H - H I , and with one new color for H I +EI, we obtain a q-coloring of H. SO q ( H ) Sq

=A

Hence, q ( H )= A(I?).

( f i ) sq ( H ) .

0

In order to include also the results of Chvatal [ 5 ] and of Fulkerson [ 131, we rephrase the theorem of Lovasz as follows: Theorem 3.1. Let H = (El,E 2 , .. . , E m )be a hypergraph on X with incidence matrix A. The following conditions are equivalent: (1) H is normal, i.e.. q ( H ' )= A ( H ' ) for every H' C H ; ( 2 ) every uertex of matching polytope Q = {y y E R m , y 3 0, Ay S 1) has (0, 1)-coordinates ; (3) every vertex of the matching polytope Q has integral coordinates ; (4) v(H;l,q)=7*(H;l,q)forevery qEN"; ( 5 ) v ( H ; I , q ) = ~ ( H ; l , qforevery ) qEN'"; (6) every partial hypergraph H' satisfies v ( H ' ) = T ( H ' ) .

I

Proof. (1) implies (2). Let z be a vertex of the polytope Q ;since z is determined by a system of equalities with integral coefficients, its coordinates are rational, and the.re exist integers k , p l , p 2,... ,pm s 0 SO that kz = ( p l , p z , . . . ,p m ) . In the hypergraph fi obtained from H by multiplying each edge E, by p i , we have

Hence, A ( H ) S k, and, by the lemma, q ( H ) S k. Thus there exists a k-coloring of the edges of fi, and each color A defines a matching GAS Put y:

=

i

1 if one copy of E, is in H A , 0 otherwise.

The vector y A = ( y ; , y:, . . . ,y i), with (0, 1)-coordinates, belongs to Q ; furthermore

Since z is a vertex of 0,

7

Minimax theorems

This shows that z is a vector with (0, 1)-coordinates.

(2) implies (3). Obvious.

I

(3) implies (4). Since max{(q, y ) y E Q } is reached by a vertex of the polytope Q, we have

(4) implies (5). Put

O,={z l z E Q , ( z , q ) = y ; ( Y , q ) ] . Since Q1is a face of the polytope Q, there exists a row-vector matrix A such that

a'l

of the incidence

z E Ql 3 ( u j l , z ) = 1. It follows from (4) that each matching of H with maximum q-values covers the vertex xjl. Put

qi - 1 if xi, E Ei,

q; =

qi

otherwise.

Thus

v(H;l,q')= v(H;l,q)-1.

As above, there exists a vertex x, of H and a vector q 2 = (q;,q:, . . . ,qL) such that v ( H ;1, q 2 )= v ( H ;1,q l ) - 1.

We continue to define a sequence a

= (xi,, x,,

. . . ,x j k ) until

we have

v(H;l,qk)=O. The vector t = (tl, tZ,. . . , t n ) , where tj is the number of appearances of x j in the sequence a, is a q-transversal of H such that " ( t , l ) = j = l tj = k = v ( H ; l , q ) . Hence t is a q-transversal of minimum value, and

4,I ) =

T(H;

V ( H ;1,q).

C.Berge

8

(5) implies (6). Let H' be a partial hypergraph of H, and put ql

={

The vector q

1 if E, E H', 0 otherwise.

= (ql, q 2 , . . ,q m ) satisfies

U(H;l,q)= V ( H ' ) ,

T(H;l,q)=T(H').

Thus (5) implies u ( H ' )= T ( H ' ) .

(6) implies (1). It suffices to show that a hypergraph H which satisfies ( 6 ) is such that q ( H )= A ( H ) . Let be a hypergraph whose vertices are the matchings of H, and where an edge E, denotes the set of all matchings of H containing E,. Clearly, E, 17 E, = 0 if and only if E, f l E, # 0. Since H satisfies (6), it has the Helly property, hence v ( H )= A ( H ) . Also,

9(H)= ~ ( f f ) , T ( G = ) q(H), d(H) = v ( H ) . Hence l? is normal; since we have already shown that (1) implies (6), we get v(z?)= T ( H ) , or, equivalently,

q (HI = A ( H ) . This achieves the proof.

0

Let G be a graph, and let A be the incidence matrix whose columns represent the vertices of G and whose rows represent the maximal cliques of G. Let S ( G ) be the set of the characteristic vectors of all the stable sets of G ; clearly, the convex hull [ S ( G ) ]of S ( G ) is contained in the matching polytope Q = {z z E R", z 3 0. Az s 1). Theorem 3.1 shows something more:

I

Corollary (Chvatal [5]). [ S ( G ) ]= Q if and only if G is a perfect graph.

Proof. Clearly, a vector s = (sl,s2,. . .,s,,) belongs to S ( G ) if and only if s is a (0, 1)-vector of Q. Consequently, the result follows from the equivalence between (1) and (2) in Theorem 3.1. 0 Another proof based on the powerful theory of antiblocking polyhedra [13] has been discovered by Fulkerson and can be found in [34].

Minimax theorems

9

4. The Menger property

A hypergraph H on X =(xI,x2, ..., x n } has the Menger property if u(H;p, 1) = 7 ( H ; p ,1) for every n-dimensional vector p with non-negative integral coordinates. Lemma. Let H = ( E l ,E 2 , .. . , E m ) be a hypergraph of order n, and let q E N". The following conditions are equivalent: (i) 7 * ( H ; p q, ) is a n integer for every p E N" ; (ii) ~ * ( H ; p , q ) = ~ ( H ; p ,for q )everypEN". Proof. Clearly, (ii) implies (i). Now we shall show that (i) implies (ii). Let H be a hypergraph with the property (i) and with incidence matrix A = ((a:)); let X be a vertex of the polytope

P, = { x I x E R " , x ~ 0 xA , aq}. We write: X E Extr(P,). It suffices to show that all the coordinates of X are integral. Let us show, for instance, that XI is integral. Put

For j < n, put

dl

=

I

C a',+l

ifjEJ.

kEK

For every x E P,, the vector d

For x

{X

= X, this

= ( d , , d2,. . . , d , )

satisfies

yields

If x E P,, x # 2, the vector x cannot be in all of the bounding hyperplanes x, = o}, ;E J, Or {X (&, X ) = qk}, k E K, SO that

I

I

Let A be a positive integer; put d ( h ) = ( A d , + 1, Adz, Ad3,. . . ,Ad,).

C. Rerge

10

Clearly, the minimum of ( d ( A ) ,x ) for x E P, is obtained for at least one vertex of the polytope P,. say x ( A ) . Since the set Extr(P,) is finite, there exists a vertex 2 E Extr(P,) which satisfies = x ( A ) for an infinity of values of A. Since

+ A I,

( d, I )

(d,X)

+A

X,

for an infinity of A's, we get

( d , i) s ( d , x). From (2) and (3), it follows that = X. Thus the minimum of the linear function ( d ( A ) ,x ) = A(d, x ) + xI is obtained for x = X with an infinity of values of A. So for some integer h, we have

h(d, 2 ) + il= min (d(h), x ) = T * ( H ;d(h), 4 ) . X€PS

From (2) and (3), we obtain also

h(d, X) = min (Ah, x) = T * ( H ;id, 4). X€Pq So X, = T * ( H ; d ( h ) , q ) -T * ( H ; h d , q ) ,and XI is an integer.

0

Theorem 4.1 (Hoffman). Let H be a hypergraph. The following conditions are equivalenr (and characterize the Menger property): (i) v ( H ;p, 1) = T ( H ;p, 1) for euery p E N " ; (ii) v ( H ; p ,1) = T * ( H ; 1) ~ ,for every p EN".

Proof. (i) implies (ii) by Proposition 2.1, and (ii) implies (i) by the lemma. 0 To recognize the hypergraphs satisfying the Menger property consider a hypergraph H = (El,E 2 , .. . , E r n )on X = {xl,x2,. . . ,x,} and an integer p, 2 1. Define a hypergraph H ( x , , p , ) obtained from H by replacing x, by a set X = {x:, xf, . . . , x?} and by replacing each edge E, containing x, by p, new edges: (E, -{~,})U{.K;}, k

=

1,2,... ,pt.

For p, = 0, define H ( x , , p , )by removing from H all the edges containing x i . We shall refer to H ( x , , p , ) as the ''multiplication'' of the vertex x, by p , . For an integral vector p = ( p , ,p 2 , .. . ,p")3 0, let H ( p )be the hypergraph obtained from H by multiplying x1 by pl, and then x2 by p z , etc. Clearly,

T(H"')= T ( H ;p, l),

v(H"') = v ( H ; p ,1).

So, if X is a family of hypergraphs H satisfying v ( H )= T ( H ) , and if H E X

Minimax theorems

11

implies H@’EX, we know that every hypergraph in X satis,fies the Menger property. Example 1. Let G be a multigraph, and let a, b be two of its vertices. Consider the hypergraph H whose vertices are the edges of G, and whose edges are the sets of edges constituting an elementary chain connecting a and b in G. In fact, the multiplication of a vertex e, of H by pt is equivalent to considering a multigraph obtained from G by replacing the edge. e, by pr parallel edges (or removing e, if p, =O). Since the Menger theorem asserts that v ( H )= T ( H ) , the multiplication principle shows that H has the Menger property. Example 2 (Edmonds [6]). Generalizing Example 1, consider a set S of vertices in G, with I S 122, and a hypergraph H whose edges are the chains connecting in G two distinct vertices of S. A theorem of Edmonds [6] asserts that v ( H ) = T ( H ) ;therefore, H has the Menger property (by the multiplication principle). Example 3. Let G be a directed graph and let a be a vertex. Consider the hypergraph H whose vertices are the arcs of G, whose edges are spanning arborescences rooted in a. A theorem of Edmonds [7] asserts that v ( H )= 7 ( H ) (see an elegant proof by LovAsz [20]). Therefore H satisfies the Menger property (by the multiplication principle). Example 4 (Schrijver [30]). Generalizing Example 3 , consider a directed graph G = ( X , U ) ,and a directed graph K = (X, V )on X such that each pair of source and sink of K is connected by a directed path of K. The family (W,, W z ,. . .) of subsets W iof U such that K + W iis strongly connected (and which are minimal) constitute a hypergraph H on U, and Schrijver [30] has proved that v ( H ) = T ( H ) .Therefore H satisfies the Menger property. This result implies, by constructing appropriate graphs K, many known results, such as Menger’s Theorem for directed graphs, Gupta’s Theorem for bipartite graphs [15], Edmonds’ branching Theorem (Example 3), a special case of a conjecture of Edmonds and Giles [8] (also proved independently by D. Younger), and a theorem of Frank [lo]. Example 5 (Rothschild and Whinston [25]). Consider a multigraph G with only even degrees, and let a, a ‘ , b, b’ be four vertices. Consider the hypergraph H whose vertices are the edges of G, and whose edges are the chains linking a and a’, or b and b‘. Rothschild and Whinston 1251 have shown that H has the Menger property.

C. Beige

12

Example 6. Let G be a simple graph, and let a and b be two vertices. Consider the hypergraph H whose vertices are the vertices of G different from a, b, and whose edges are the chains linking a and b in G. By the theorem of Menger and by the multiplication principle, H satisfies the Menger property. An equivalent way to study the Menger property is to consider the transversal hypergraphs. For a hypergraph H on X = {xl,x2,. . . ,x n } , and for a non-negative integer-valued vector p = ( p , ,p 2 , .. . ,p"), let H P denote the hypergraph obtained from H by replacing each vertex x, by a set X , with IX, I = p,, and by replacing each edge E, by u { X , Ix, EE,}. Theorem 4.2. Let H be a hypergraph on X = { x I , x 2 ,..., x,,}, and let p ( p l , p 2 ,... ,p n )E N " . Then Tr[H'"]

= [Tr HI",

Tr[H"]

=

= [Tr HI"'.

The proof is easy. Let a ( H ) denote the maximum number of colors needed to color the vertices of H ss that every edge contains all the colors. Clearly cr(H)sminIE,I = s ( H ) . We shall say that H satisfies the Gupta property if a ( H P )= s(HP)for every p E N " . For instance, if G is a bipartite multigraph, the dual G * satisfies the Gupta property by a theorem of Gupta [ 151.

Theorem 4.3. A hypergraph satisfies the Gupta property if and only if its transversal hypergraph satisfies the Menger property. Proof. By Theorem 4.2, a ( ~= v~[ T r)( H P ) = ] V[(T~H)'~'],

s ( H ~=)T [ T ~ ( H =~ T) [] ( T ~ H ) " ' ] . Hence the result follows.

5. Paranormal hypergraphs A hypergraph H of order n is paranormal if 7 ( H ; p ,1)= 7 * ( H ; p ,1) for every n-dimensional vector p 3 0. By Proposition 2.1, every hypergraph with the

Minimax theorems

13

Menger property is paranormal. The converse is not true. For instance, the dual K : of the 4-clique K4 is paranormal, but v(K:)= 1 and T ( K $ = ) 2; hence K $ does not have the Menger property.

Proposition 5.1. Let H be a hypergraph on X = {x,,x 2 , . . . ,x,} with incidence matrix A. The following conditions are equivalent (and characterize the paranormal hypergaphs): (i) ~ ( H ; p , l ) = ~ * ( H ; pfor , l )a l l p E N " ; (ii) T * ( H ; ~1 ), is integral for all p E N " ; (iii) every vertex of the transversal polytope P = { t t E R ', t > 0, tA 3 1) has (0, 1)-coordinates.

1

Proof. This follows immediately from Theorem 4.1 and its lemma.

0

Several examples of paranormal hypergraphs arise in matroid theory (see Woodall [35]).Other examples follow from the theory of T-joins developed by Edmonds and Johnson [9] to generalize the famous Chinese Postman problem of Guan Megu. Let G = (X, E ) be a multigraph without loops, and let T be a non-empty subset of X . A T-join is a minimal subset F C E such that the partial graph G' = (X, F ) satisfies

I

{ x d&)=

1 modulo 2} = T.

A T-cut is an elementary cocycle w ( A ) , A C X, such that

/ A n TI = 1 modulo 2. It is easy to show that the transversal hypergraph of the T-join hypergraph is the T-cut hypergraph, and vice versa. Edmonds and Johnson have proved that every T-join hypergraph is paranormal (see also Seymour [31]). Example 1. Let G be a simple connected graph on X where I X I is even. An X-join of G is a spanning forest with only odd degrees. An X-cut is a cocycle w ( A ) where [ A I is odd. Example 2. Let a, b be two vertices of G. Let T = {a, b } . A T-join is an elementary chain connecting a and b. A T-cut is a cut between the vertices a and b.

I

Example 3. Let T = { x x E X, dG( x )3 1 modulo 2). A T-join is a minimal set of edges which need to be duplicated to get an eulerian graph. For A C X ,

Hence I o ( A ) Jis odd if and only if I T n A I is odd. So a T-cut is an elementary cocycle w ( A ) with an odd number of edges.

6. Balanced hypergraphs

Let H = ( E l ,E 2 , .. . ,E m )be a hypergraph with incidence matrix A. A cycle is a sequence (xI1,E,,,xQ,E,, . . .,xIk,E,,, x,J, where k 2 2, the x,,,’s are distinct vertices, the E p ’ s are distinct edges, and each edge Elpcontains the two vertices which are before and after it in the sequence. A hypergraph H is balanced if every cycle with an odd number of edges admits an edge containing at least three vertices of the sequence - or, equivalently, if the incidence matrix A does not contain an odd square submatrix of the the following type: 1 1 0 0 . .

. 0 0

0 0 0 0 . . . 1 1 1 0 0 0 . . . 0 1 We introduced this concept to extend theorems about hypergraphs with no odd cycles and totally unimodular matrices. Proposition 6.1. Let H be a balanced hypergraph ; then every partial hypergraph H’ is also balanced. Proof. See [l].

0

Proposition 6.2. Let H be a balanced hypergraph on X, and let S C X ; then the subhypergraph Hs induced by S i s also balanced. Proof. See [l]. Proposition 6.3. Let H be a balanced hypergraph, then its dual H * is also balanced. Proof. See [l].

0

Minimax theorems

Proposition 6.4. Let H be a balanced hypergraph on X is also balanced, p E N".

15 = {xl, x2,. .

. ,x"}; then H P

Proof. It suffices to prove the result for p = (0,1,1,. . . ,1) and for p = (2,1,1,. . . ,l). In the first case, H P = Ha, where A = X - {xl}, and the result follows from Proposition 6.2. In the second case, H P is obtained from H by replacing x , by a set X1= {xi, x;}. Let p be an odd cycle of H P which does not have an edge containing three vertices of the sequences. If p contains both xl and x;, then p = ( . .., z,E,xl, ...,xY) and the edge E which precedes x ; contains the vertices z, x i and x;, which is a contradiction. If p does not contain x;, say, then p induces on H an odd cycle with no edge containing three vertices, which is a contradiction. 0 Proposition 6.5. A hypergraph H is balanced if and only if every induced subhypergraph Hs is bicolorable (i.e., there exists a bipartition (S1, S2) of S such that every edge of Hs with more than one point meets both S1 and S,). Proof. See [l].

0

Proposition 6.6. Let H be a balanced hypergraph on X H @ )is also balanced, p E N " .

= { x l ,x2,. . . ,x,,};

then

Proof. It suffices to prove the result for p = (0,1,1,. . . , 1 ) and p = (2,1,1,. . . , 1). In the first case, the result follows from Proposition 6.1. In the second case, H(") is obtained from H by replacing x 1 by two additional vertices x I and xY, and each edge E containing x1 by two edges E ' = ( E -{xl})U{xl} and ( E - {XJ) u ( x 9 . Every bicoloring of H induces a bicoloring of H ( p (by ) giving the same color to x I and x:'), and every bicoloring of H @ in ) which x i and x; have the same color induces a bicoloring of H. So, if an induced subhypergraph K of H @ )has no bicoloring, then K contains both x i and x;' and K is induced by a subhypergraph Hs. Since Hs has a bicoloring, the contradiction follows. 0 El'=

Proposition 6.7. A hypergraph H is balanced if and only if q (Hk) = A (Hk) for every partial subhypergraph H;. Proof. See [l]. Proposition 6.8. A hypergraph H is balanced if and only if every induced subhypergraph Hs is normal.

C. Berge

16

Proof. This follows from Proposition 6.7 and the definition of normality.

0

Proposition 6.9. A hypergraph H is balanced if and only if every partial subhypergraph HL satisfies the Konig property

v(HL)= 7(Hk). Proof. This follows immediately from Proposition 6.8 and Theorem 3.1. 0 The first proof of this result was due to Berge and Las Vergnas [3], but a simpler proof was found later by LovBsz.

Proposition 6.10. A hypergraph H is balanced if and only if

v ( H s ; I , q ) = T ( H s ; l , q ) ( S C X , qEN"'). Proof. From Proposition 6.8 and Theorem 3.1. 0 Proposition 6.11. A hypergraph H is balanced if and only if

v ( H s ; l , q ) = ~ * ( H s ; l , q () S C X , q E N " ' ) . Proof. From Proposition 6.8 and Theorem 3.1. 0

Proposition 6.12. A hypergraph H is balanced i f and only i f every partial hypergraph H' has the Menger property. Proof. Let H be a balanced hypergraph on X = {xl,x2,. . . ,x.} and let p E N " . By Proposition 6.6, H'p' is balanced. By Proposition 6.9,

v ( H ;p. 1) = v(H"') = T ( H ( ~=)T) ( H ; 1~) ., So every partial hypergraph H' C H also satisfies v ( H ' ;p, 1 ) = T ( H ' ;p, 1). Hence H' has the Menger property. Conversely, let H be a hypergraph whose partial hypergraphs have the Menger property. Assume that H is not balanced; there exists an odd cycle, say (xl,El,xZ.E2,. . . ,X 2 k + l r EZk+l, x,), with no edge containing three vertices. Put H' = (El,E2,.. . , Ezk+Jand p, = 1 if i S 2 k + 1 , p , = + r: if 2 k + 1 < i 6 n. We have ~ ,= k + 1. v ( H ' ;p, 1) = k, T ( H ' ; 1) So H' does not have the Menger property, which is a contradiction.

0

The fact that every balanced hypergraph has the Menger property was first proved (by different methods) by Fulkerson, Hoffman and Oppenheim [14].

Minimax theorems

17

Proposition 6.13. A hypergraph H is balanced if and only if every partial hypergraph H’ is paranormal. Proof. Let H be a balanced hypergraph. By Proposition 6.12, H’ is paranormal. Now, let H be a hypergraph whose partial hypergraphs are paranormal. There exists an odd cycle (XI, E l , . . . ,EZk+1,XI) with no E, containing three x,’s. Define (as above) H’ and p. Then T ( H ‘ p, ; 1)=k

+ 1,

T * ( H ‘ p, ; 1 ) = ( 2 k + 1)/2.

Hence by Proposition 5.1, H’ is not paranormal, which is a contradiction.

0

Proposition 6.14. A hypergraph is balanced if and only if every partial hypergraph has the Gupta property. Proof. As above, it suffices to show that a balanced hypergraph H of order n satisfies the Gupta property. Since H P is also balanced for every p E N ” (Proposition 6.4), it suffices to show that a ( H )= s ( H ) . Let k = min I E, 1, and let (SI,Sz,. . . ,s k ) be a partition of the vertex-set X into k classes. Denote by k ( i ) the number of classes which meet E,. If k ( i ) = k for all i, all the classes are transversal sets of H, and the proof is achieved; otherwise, there exists an index j S m with k ( j ) < k. Since k ( j )< k c ( E ,1, there exists a class S, such that

Is, n~,I==2. Furthermore, there exists a class S, such that

1 S, f l E, 1 = 0. The subhypergraph HspuSqis balanced, and therefore has a bicoloring (Sb, S:). Put S : = S, for r# p, q. The partition (Sl, S : , . . . ,S : ) determines as above new coefficients k ’ ( i ) , and

k ’ ( i ) * k ( i ) ( i S m ) ; k ’ ( j ) = k ( j ) + 1. With this method, it is always possible to improve a partition (Sr, S:, . . . , S ; ) until we have k ’ ( i )= k for all i ; then we have a partition of X into k = s ( H ) transversal sets, and the proof is achieved. 0

Proposition 6.15. Let H be a balanced hypergraph. Then the transversal hypergraph Tr H has the Menger property. Proof. This follows from Proposition 6.14 and Theorem 4.3. 0

C. Berge

18

Note that this result is an answer to one open problem raised by Fulkerson, Hoffman and Oppenheim [14], who found a balanced hypergraph whose transversal hypergraph is not balanced, and asked if such a transversal hypergraph has similar properties. The above results can be applied to the class of graphs defined in the introduction, and to many hypergraphs arising in graph theory. In particular, every hypergraph whose incident matrix is totally unimodular is balanced (because this class is characterized by the Ghouila-Houri property: every subhypergraph admits a bicoloring which splits every edge in two equal parts). For instance, the dual of a bipartite multigraph is balanced. Also, if T is an arborescence on X , and if H is a hypergraph on X whose edges E, are subsets of X such that TE,is a path, it is clear that H is unimodular and, consequently, balanced. Other balanced hypergraphs have been considered by Frank [12]. We can also summarize the above results by considering the property v ( H ; p , q ) = T ( H ; 4). ~ , This equality holds for every p E N " and every q E N " if and only if h is a unimodular hypergraph. If H is a balanced hypergraph, Proposition 6.10 states that the above equality holds for every p E N " with (1. + 1;)-coordinates, and every q E N". Also, Proposition 6.12 states that the above equality holds for every p E N", and every q E N " with (0, 1)-coordinates. But the above equality does not hold for every p E N" and every 4 E N". For instance, let H be a hypergraph with edges E , = {1,4}, E , = {2,4}, E , = {3,4}, E , = {1,2,3,4}.Let p = (2,2,2,3), q = (1,1,1,2). It is easy to see that t = ( I~ ,I 2 I, 2 I, ~ ) belongs to the q-transversal polytope of H, and z = (f,f,f,i)belongs to the p-matching polytope. Since ( p , 1 ) = (q,z ) = 4.5, we have (by Proposition 2.1) T * ( H ; q~ ), = 4.5. Hence V(ff;p,

4 ) = 4 < 4.5 = T * ( H ; p ,q ) < 5 = T ( H ; p ,4 ) .

We do not know what are all the pairs (p,q ) such that v ( H ;p , q) = T ( H ;p , q ) for every balanced hypergraph H with n vertices and m edges, i.e., if there is a common generalization to Proposition 6.10 and Proposition 6.12. References [I] C. Berge, Balanced matrices, Math. Program. 2 (1972) 19-31. [2] C. Berge. Balanced matrices and property G , Math. Program. Study 12 (1980) 16>175. (31 C . Berge, C.C. Chein, V. Chvital and C.S. Seow, Combinatorial properties of polyominoes, Combinatorica 1 (1981) 217-224. [4] C. Berge and M. Las Vergnas, Sur un thCortme du type Konig pour hypergraphes, Ann. N.Y. Acad. Sci. 175 (1970) 32-40. [S] V. Chvatal, On certain polytopes associated with graphs, J. Comb. Theory, Ser. R, 18 (1975) 13RlS4. [ 6 ] J. Edmonds, Submodular functions, matroids and certain polyhedra, Combinatorial Structures and their Applications (Gordon and Breach, 1969) 6%87.

Minimax theorems

19

[7] J. Edmonds, Edge disjoint branchings, Combinatorial Algorithms, New York, 1973, 91-96. [8] J. Edmonds and R. Giles, A min-max relation for submodular functions on graphs, Studies in Integer Programming, Ann. Discrete Math. 1 (1977) 185-204. [9] J. Edmonds and E.L. Johnson, Matching, Euler Tours and the Chinese Postman, Math. Program. 5 (1973) 88-124. [lo] A. Frank, Kernel systems of directed graphs, Acta Sci. Math. (Szeged) 4 (1979) 63-76. [ l l ] A. Frank, Covering branchings, mimeo, Janos Bolyai Institute, 1979. [12] A. Frank, On a class of balanced hypergraphs, mimeo, Research Inst. for Telecom., Budapest, 1979. [13] D.R. Fulkerson, Blocking and anti-blocking pairs of polyhedra, Math. Program. 1 (1071) 16&194. [I41 D.R. Fulkerson, A.J. Hoffman and R. Oppenheim, On balanced matrices, Math. Program. Study 1 (North-Holland, Amsterdam, 1974) 120-132. [15] R.P. Gupta, An edge-coloration theorem for bipartite graphs, Discrete Math. 23 (1978) 229-233. [16] A.J. Hoffman, A generalization of max-flow min-cut, Math. Program. 6 (1974) 352-359. [17] A. Lehman, On the width-length inequality, mimeo, 1965. [18] L. LovLsz, 2-matchings and 2-covers of hypergraphs, Actes Math. Acad. Sci. Hung. 26 (1975) 433444. [19] L. Lovisz, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-267 (this volume, pp. 29-42). [20] L. LovLsz, On two min-max theorems in graphs, J. Comb. Theory, Ser. B, 21 (1976)96103. [21] J. F. Maurras, Polytopes i sommets dans [O, l]“, These doctorat d’etat, Paris VII, 1976. [22] M.W. Padberg, On the facial structure of set packing polyhedra, Math. Program. 5 (1973) 199-215. [23] M.W. Padberg, Perfect zero-one matrices, Math. Program. 6 (1974) 18&196. [24] M.W. Padberg, Almost integral polyhedra related to certain combinatorial optimization problems, Linear Algebra & Appl. 15 (1976) 69-88. [25] B. Rothschild and A. Whinston, On 2-commodity network flows, Oper. Res. 14 (1966) 377-387. [26] M. Sakarovitch, Quasi-balanced matrices, Math. Program. 8 (1975) 382-386. [27] M. Sakarovitch, Sur quelques problemes d’optimisation combinatoire, These doctorat d’etat. Grenoble, 1975. [28] A. Schrijver, Fractional packing and covering, Packing and Covering (Sti. Math. Centrum Amsterdam, 1978) 175-248. [29] A. Schrijver and P.D. Seymour, A proof of total dual integrability of matching polyhedra, Math. Centrum report ZN, 1977. [30] A. Schrijver, Min-max relations for directed graphs, Report AE 21/80, University of Amsterdam, 1980. [31] P.D. Seymour, On multicolorings of cubic graphs and conjectures of Fulkerson and Tutte, mimeo, Oxford, 1977. [32] P.D. Seymour, The forbidden minors of binary clutters, J. London Math. Soc. 12 (1976) 356-360. [33] P.D. Seymour, Discrete Optimization, Lectures Notes, Univ. Oxford, 1977. [34] L. E. Trotter, Solution characteristics and algorithms for vertex packing problems, Thesis, Cornell University, 1973. [35] D.R. Woodall, Menger and Konig systems, Theory and Applications of Graphs; Springer Verlag Lecture Notes in Math. 642 (1978) 620-635.



A CLASS OF BICHROMATIC HYPERGRAPHS Jean-Claude FOURNIER Universitt Paris-Val de Maine, U.E.R. de Sciences Economiques et de Gestion, 94210 La Varenne St-Hilaire, France

Michel LAS VERGNAS* Universitt Pierre el Marie Curie, U.E.R. 48 - Mathtmatiques, 7.5005 Paris, France

We give a sufficient condition for bichromatic hypergraphs in terms of properties of cycles. Application: The set of inclusion-maximal cliques of a perfect graph can be partitioned into two classes such that both classes are represented at every vertex contained in at least two inclusion-maximal cliques.

1. Bichromatic hypergraphs and perfect graphs

Let G be a graph with vertex-set V. A clique of G is a subset of V inducing a complete subgraph of G. We denote by K ( G ) the hypergraph with vertex-set V whose edges are inclusion-maximal cliques of G. We recall that a hypergraph H is normal ([l],[ 5 ] )if v ( H ’ ) =7 ( H ‘ )for all partial hypergraphs H‘ of H, where v(H’) is the maximal number of pairwise disjoint edges of H’ and T ( H ’ )is the minimal cardinality of a set of vertices meeting all edges of H ’ . A partial hypergraph H’ of H is constituted by a subset of the edge-set of H [l]. It is easily seen that a graph G is perfect if and only if K ( G ) * , the dual hypergraph of K ( G ) ,is normal. By definition of a dual hypergraph [l], vertices of K ( G ) * are inclusion-maximal cliques of G and edges of K ( G ) * are all sets K,(G),x E V, where Kx(G) is the set of inclusion-maximal cliques of G containing the vertex x. A hypergraph H = ( X , ‘8) is bichromatic if there is a partition X = B + R , called a bicoloring, such that every edge E E 8 with lE 122 meets both B and R. In relation with his work on perfect graphs Lovasz asked: Is every normal hypergraph bichromatic? [5]. We answered positively this question in the following more general form:

Theorem 1 [2]. A hypergraph with no odd cycles of maximal degree two is bichromatic. * C.N.R.S. 21

22

3.-C.Foumier, M. Las Vergnas

In a hypergraph a cycle (with length k ) is defined by a sequence ( x l .E l ,x2. E2,. . . ,X k , E k , x l )of pairwise distinct vertices x I ,xZ,. . . ,Xk and pairn E, for i = 2,3,. . . , k and wise distinct edges El,E z , .. . ,E, such that x, € E,-l xI E Ek f l El. A cycle is of maximal degree two if any three of its edges have an empty intersection. An odd cycle is a cycle with an odd length k 2 3 . Clearly a normal hypergraph contains no odd cycles of maximal degree two. Hence by Theorem 1 every normal hypergraph is bichromatic. Equivalently we have the following corollary:

Corollary. The set of inclusion-maximal cliques of a perfect graph can be partitioned into two classes such that both classes are represented at each vertex contained in at least two inclusion -maximal cliques. Theorem 1 can be equivalently stated: A hypergraph with no odd cycles of maximal degree two such that any two non-consective edges of the cycle are disjoint is bichromatic (see [2]). A conjecture due to Sterboul [6] proposes a strengthening of this form of Theorem 1:

Conjecture [6]. A hypergraph with no odd cycles ( x , , El,x Z ,E 2 , .. . ,x k , Ek,x , ) such that any two non-consecutive edges are disjoint and I E, f l E,+l1 = 1 for i = 1 , 2 , . . . ,k - 1 is bichromatic. Clearly a cycle with length > 3 such that any two non-consecutive edges are disjoint is of maximal degree two. O n the other hand a 3-cycle ( x , , El,xZ, EZ, x , , E 3 ,x l ) such that I El r l E,I = I E2 fl Ejl = 1 is also of maximal degree two. Hence Sterboul’s conjecture implies Theorem 1. We emphasize that in Sterboul’s conjecture the condition 1 El fl Ek I = 1 is not required. Actually there are non-bichromatic hypergraphs containing no odd cycles such that any two non-consecutive edges are disjoint and any two consecutive edges have exactly one common vertex (for example, the collection of ( n + 1)-subsets of a set with 2 n + 1 elements). In other words Sterboul’s conjecture is not true if we require I El fl Ek I = 1. In the present note we prove a theorem intermediate between Sterboul’s conjecture and Theorem 1. The main results were first published in French by the same authors in [3]. All hypergraphs considered in this paper are finite and without multiple edges.

2. A sufficient condition for bichromatic hypergraphs

Theorem 2. A hypergraph with no odd cycles

(XI,

EI,X Z , E2,.. . ,x k , Ek, X I )

Of

A class of bichromatic hypergraphs

maximal degree two such that IE, n E,+l\= 1 for i bichroma tic.

23 = 1,2,. . . , k

-

1, is

The following lemma immediately implies Theorem 2 by induction on the number of vertices:

Lemma. Let H = ( X , 8 ) be a hypergraph containing an edge EoE 8, I Eo1 3 2 , such that H \Eo = ( X , % \{E,,}) is bichromatic. Suppose further that there is a vertex z E Eo such that every odd cycle (xl, E l ,xz,E2,.. . , xk, Ek, x l ) with maximal degree two satisfying 1 E, n E,+I1 = 1 for i = 1,2,. . . ,k - 1 contains Eo, suy E l = Eo, and has x i , xz # z. Then H is bichromatic. Proof. Let X = B + R (blue and red) be a bicoloring co of H\Eo. We assume notations such that EoC B (if E , meets both B and R there is nothing to prove). The central idea of the proof is to construct inductively a sequence co,cl, . . . , c, of bipartitions of X , c,: X = B, + R,, by interchanging colors blue and red on a set meeting all edges of H monochromatic in c,-,. Thus at each step the edges not bicolored by c , - ~become bicolored by c,. The hypothesis of the lemma ensures that this algorithm does not cycle: Let 8, denote the set of edges monochromatic in c,. We will prove that the 8,’s are pairwise disjoint. Hence by finiteness %, = B for some p , i.e., c, is a bicoloring of H. More precisely: At the beginning go = (Eo}.We set To= ( z } and define c1 by B1= B \ Toand R I = R To.In general suppose c , - ~has been defined, i 3 2. We have E , g 8,-1(the set of edges monochromatic in c , - ~ ) ;this fundamental property will be proven inductively. Hence B, resp. R, meets all edges in 8,-l. For i even, resp. i odd, let T,-, be a set contained in R, resp. in B, meeting all edges in and minimal with respect to inclusion with these properties. We define c, by B, = B,_,+T-,, R, = R,-,\T,-, if i is even, and B, = B,-l\T,-,, + T,-I if i is odd. R, = We prove the following properties by induction on p 2 1: ( 1 . p ) gP is disjoint from go, ,..., 8p-i; ( 2 . p ) every edge in gp meets Tv-land is blue in c, (i.e., contained in B,,) for p even, and red for p odd; (3.p) T, is disjoint from Eo, T 1 , .., . Tp-l. The case p = 1 is immediate. Suppose p 3 2 and ( l . i ) , (2.i), (3.i) hold for i = 1 , 2 ,..., p - 1 .

+

Proof of (1.p). An edge E E 8,, 1 C i G p - 1, meets T,-I by (1.i) and meets also T, by definition of T,.Now by (3.1), . . . ,( 3 . p - 1) the sets Eo, T I , .. . , T+ are pairwise disjoint, hence all T, with an even index i < p - 1 are red in c, and all T with an odd index are blue. It follows that all edges in g1,g 2 ,..., are

J.-C. Fournier, M. Lm Vergnas

24

bicolored in c,. O n the other hand, Eo is also bicolored since Eo\T , is blue in c, and T , is red. Hence gP is disjoint from 8", 8,,. . . , by (1.p). Since E is Proof of (2.p). An edge E E 8 ' , is bicolored in monochromatic in c,,, necessarily E meets Tp-Iand its color is that of Tp-lin c,, hence (2.p).

Proof of ( 3 . p ) . It is easily seen that T, is disjoint from To,T I , .. . , Tp-l. Suppose for instance p even. The edges in E, are blue in c, by (2.p) and T,, T z , ... , Tp-2 are red by (3.1),. . . ,(3.p - 1). Hence T, does not meet T,, T2,.... Tp-2.On the other hand T, is contained in B and T I ,T 3 , .. . ,T,-, in R . The case p odd is analogous. We have thus to prove that T, n (&\ To)= 0. This is clear for p odd since Eo C B and T, C R in this case. Suppose p is even. We show that E,, fl E = 0 for all edges E E gP. Let E, E 8, and suppose for a contradiction that there is xo E E, f l ( E o \ { z } ) . Let x, E Tp-lf l E,. By the minimality of T,-, there is Ep-lE such that E,-, fl T,-, = {x,}. Since is red in c,-~ by ( 2 . p - I), and €, is blue in c, by (2.p) we have n E, Tp-I,hence E,-, n E, C EP-,n Tn-,= { x p } , and thus E,-, n E, = { x , } . Repeating this procedure with instead of E,, and inductively, we get a sequence E P , ~ P , E p -,,. l r.~. ,xI P = z, Eo with E, E g,, x, E 7'-, and E, n = {x,-J of pairwise distinct edges and pairwise distinct vertices, hence an odd cycle (xu, Eo,x I , E l , .. . ,x,, E,, xo). By the hypothesis of the lemma this cycle is not of maximal degree two there are three edges E,, El, E,, 0 s i < j < k G p, such that E, f l El n E, # @. Suppose i, j , k are chosen with this property such that k - i is minimal. Let y E E, n El n €5. We first show that y # x,+,, xlrZ,. . . ,xk. Suppose k is even. The edge E, being blue in c k by ( l . k ) , y is different from xl, x s r . ..,& - I which are red in c, by (3.1), . . . ,( 3 . k - 1). Suppose y = x,, s even, i + 1 S s S k. If s < k the cycle (xs,Es,xs+l,E,,,,. . . ,X k , & , x s ) is an odd cycle with maximal degree two (by the minimality of k - i ) contradicting the hypothesis of the lemma. Hence s = k . Necessarily i and j are odd - if i is even, i7 being blue in c, cannot contain xk which is red in c,. But then the cycle ( x k , E,,x , + ~E,+I,. , . . ,x,, El, & ) is odd with maximal degree two, contradicting the hypothesis of the lemma. The case k odd is similar. The indices i and j have necessarily different parities - otherwise the cycle (y, E,, x,+,,E,,,, . . . ,x,, E l , y ) is odd with maximal degree two, contradicting the hypothesis of the lemma. Similarly j and k have different parities. A final contradiction arises as follows: suppose i even, j odd, k even (the case

c

A class of bichromatic hypergraphs

25

i odd, j even, k odd is similar). Since E, is blue in c, and El n R is red in c,, we have E, n El C B. Similarly El n EkC R. Therefore E, n El fl El, = 0. 0 Theorem 2 implies Sterboul’s conjecture for hypergraphs with 2- or 3-element edges: Corollary. Let H be a hypergraph with rank S 3 containing no odd cycle (xl, El,x2, Ez,. . . ,X k , E k ,xl) such that any two non-consecutive edges are disjoint 1 = 1 for i = 1,2,. . . ,k - 1. Then H is bichromatic. and 1 E, n E,+I

Proof. We show that H satisfies the hypothesis of Theorem 2. Suppose, for a contradiction, that there is an odd cycle (xl, E l , x2, E 2 , .. . ,x k , Ek, x l ) with maximal degree two such that I E, n E,+,I = 1 for i = 1,2,. . . , k - 1. We consider such a cycle with minimal length k. By hypothesis there are two non-consecutive edges, say El and El, 3 S j S k - 1, such that I El f El l I # 0. Let y E El n El ; y is different from xl, x 2 , . . . , X I , since the maximal degree is two. As I El 1, IE, I s 3 necessarily El = { y , xl, x2}, El = {y, x,, x,+J and El n El = {y}. Hence one of the two cycles (xl,El, y, El, x,+I, EI+2,x,+2,. . . ,X k , Ek, X I ) and (y,E1,x2,E2,x,,...,x,, E,, y ) is odd and thus contradicts the minimality of k. 0 This proof still works if H contains a unique edge with more than three vertices. However, if H contains several edges with cardinality 3 4 the above proof cannot be used, as an odd cycle (xl, E l , x2,. . . ,x k , E k ,xl) with maximal degree two such that 1 Ein Ei+,I = 1 may not contain any odd cycle with this property such that, furthermore, non-consecutive edges are disjoint. An example is the hypergraph with edges 01, 067, 1289, 23, 34, 4589, 567.

3. Minimal non-bichromatic hypergraphs A hypergraph H = ( X , 8)is minimal non-bichromatic if H is non-bichromatic and H \ E is bichromatic for every E E 8. Clearly a hypergraph H = ( X , ‘i9) is bichromatic if and only if there is no partial hypergraph of H which is minimal non-bichromatic. Theorem 1 and Theorem 2 have the following equivalent statements as properties of minimal non-bichromatic hypergraphs:

Theorem 1. In a minimal non-bichromatic hypergraph there is an odd cycle with maximal degree two such that any two non-consecutive edges are disjoint. Theorem 2. In a minimal non-bichromatic hypergraph there is an odd cycle with

26

J.-C. Foumier, M. Las Vergnas

maximal degree two such that, with at most one exception, all intersections of two consecutive edges are of cardinality one. This form of Theorem 1 has the following strengthening:

Theorem 1' ([4]). Let H = ( X , a ) be a minimal non-bichromatic hypergraph. For every E E % and z E E there is an odd cycle ( x l , E l = E x 2 = z,E2,..., xk,Ek,xl) with maximal degree two such that any two non-consecutive edges are disjoint and

E ,n E~ = { z } . The corresponding statement for Theorem 2 does not hold: The hypergraph with edges 12, 179,235, 34,356, 45,679, 78,89 is minimal non-bichromatic. The edge 12 is not contained in any odd cycle with maximal degree two. We have the following theorem:

Theorem 3. In a minimal non-bichromatic hypergraph which is not a graph ( o d d )cycle there are at least two different odd cycles with maximal degree two such that, with at most one exception, all intersections of two consecutive edges are of cardinality one.

Proof. Let H be a minimal non-bichromatic hypergraph, different from a graph (odd) cycle. By Theorem 2, H contains an odd cycle ( x l ,El,x2,E?,. . . ,x k , Ek, x,) with maximal degree two such that IE, f l E,,, I = 1 for i = 1,2,. . . , k - 1. Suppose there is no other such cycle. If IElnEk1 3 2 we set E o = E l ; let z E El n Ek\ { x l } ,I f ] E lf l Ek I = 1, since H is not a graph there is an edge E, with IE, I > 3. We set En = E, ; let z E E, \{x,, x , + ~}Note . that z is in no other edge of the cycle (otherwise we could form a second odd cycle with the required properties). In both cases, for these choices of En and z the hypergraph H satisfies the hypothesis of the lemma of Theorem 2. Hence H is bichromatic, a contradiction. 0 The number two in Theorem 3 is best possible: in the above example of a minimal non-bichromatic hypergraph there are only two odd cycles with the required properties.

Remark. The proof of Theorem 2 is constructive and provides a polynomial algorithm for constructing either a bicoloring of a hypergraph or an odd cycle with maximal degree two such that, with at most one exception, all intersections of two consecutive edges are of cardinality one. A rough estimate of the complexity of this algorithm is in O ( m z n ' ) ,where m is the number of edges of the hypergraph and n its number of vertices.

A class of bichromaric hypergraphs

27

References [ I ] C . Berge, Graphes et hypergraphes, 26me i d . (Dunod, Paris, 1973). [2] J-C. Fournier et M. Las Vergnas, Une classe d’hypergraphes bichromatiques, Discrete Math. 2 (1972) 407-410. [3] J-C. Fournier et M. Las Vergnas, Une classe d’hypergraphes bichromatiques 11, Discrete Math. 7 (1974) 99-106. [4] M. Las Vergnas, Sur les hypergraphes bichromatiques, Hypergraph Seminar, C. Berge and D.K. Ray-Chaudhuri, eds., Springer Lecture Notes in Math. No. 411 (Springer, Berlin, 1974) 102-1 10. [5] L. Lovasz, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-267 (this volume, pp. 29-42). [6] F. Sterboul, Communication at the Graph Theory Seminar, Paris, 1973.


Annals of Discrete Mathematics 21 (1984) 29-42 Science Publishers B.V.

0 Elsevier

NORMAL HYPERGRAPHS AND THE WEAK PERFECT GRAPH CONJECTURE* L. LOVASZ Mathematical Institute, Eotvos L. University, H. 1088 Budapest, Hungary A hypergraph is called normal if the chromatic index of any partial hypergraph H' of it coincides with the maximum valency in H'. It is proved that a hypergraph is normal iff the maximum number of disjoint hyperedges coincides with the minimum number of vertices representing the hyperedges in each partial hypergraph of it. This theorem implies the following conjecture of Berge: The complement of a perfect graph is perfect. A new proof is given for a related theorem of Berge and Las Vergnas. Finally, the results are applied o n a problem of integer valued linear programming, slightly sharpening some results of Fulkerson.

Introduction Let G be a finite graph and let x ( G ) and w ( G )denote its chromatic number and the maximum number of vertices forming a clique in G, respectively. Obviously, (11

x ( G ) s@(GIThere are several classes of graphs such that

x ( G )= w ( G ) ,

(21

e.g., bipartite graphs, their line graphs and complements, interval graphs, transitively orientable graphs, etc. Obviously, relation (2) does not say too much about the structure of G ; e.g., adding a sufficiently large clique to an arbitrary graph, the arising graph satisfies (1). Berge [l, 21 has introduced the following concept: a graph is perfect ( y perfect) if the equality holds in (2) for every induced subgraph of it. The mentioned special classes of graphs have this property, since every induced subgraph of them belongs to the same class. H e formulated the following two conjectures in connection with this notion:

Conjecture 1. A graph is perfect if and only if neither itself nor its complement contains an odd circuit without diagonals. * Reprinted from Discrete Math. 2 (1972) 253-267. 29

30

L. Looas2

Conjecture 2. Let a ( G )denote the stability number of G, let 6(G) denote the minimum number of cliques which partition the set of all the vertices. A graph G is perfect if and only if a ( G ' )= 6(G') for any induced subgraph G' of G. This conjecture is an attempt to explain some similarities between the properties of the chromatic number and the stability number; his next conjecture is proved in the present paper, formulated as follows.

Perfect Graph Theorem. The coniplemeitt of

c1

perfect graph is perfect as well.

Obviously, thc second conjecture of Berge would follow from the first one. However, due to its simpler form, it has more interesting applications and has been more investigated. Partial results are due to Berge [ 3 ] , Berge and Las Vergnas [4], Sachs and OIaru (61. Fulkerson [S] reduced the problem to the following conjecture, using the theory of anti-blocking polyhedra:

Conjecture 3. Duplicating an arbitrary vertex of a perfect graph and joining the obtained two vertices by an edge, the arising graph is perfect. In 91 we prove a theorem which contains Conjecture 3 . Berge has observed that the perfect graph conjecture has an equivalent in hypergraph theory, interesting for its own sake too. The correspondence between graphs and hypergraphs is simple and enables us to translate proofs formulated in terms of graphs into proofs with hypergraphs, and conversely. In $2 we deduce the hypergraph version of the perfect graph theorem from the above-mentioned conjecture of Fulkerson; the proof is short and docs not use the theory of anti-blocking polyhedra. It could be formulated in terms of graphs as well; however, the hypergraph version shows the idea more clearly. I t should be pointed out that thus the proof consists of two steps and the more difficult second step was first carried out by Fulkerson. In $3, wc give a new proof of a related theorem of Berge. Finally, in $4 we give some formulations of the results in terms of linear programming. Most of them have been observed to be equivalent to the perfect graph theorem already proved by Fulkerson.

1.

Let G, H be two vertex-disjoint graphs and let x be a vertex of G. By substituting H for x we mean deleting x and joining every vertex of H to those vertices of G which have been adjacent with x.

Normal hypergraphs and the Weak Perfect Graph Conjecture

31

Theorem 1. Substituting perfect graphs for some vertices of a perfect graph the obtained graph is also perfect. Proof. We may assume that only one perfect graph H is substituted for a vertex x of a perfect graph G. Let G’ be the resulting graph. It is enough to show that

since for the induced subgraphs of G’, which arise by the same construction from perfect graphs, this follows similarly. We use induction on k = w ( G ‘ ) . For k = 1 the statement is obvious. Assume k > 1. It is enough to find a stable set T of G‘ meeting all k-element cliques, since then coloring these vertices by the same color and the remaining vertices by k - 1 other colors (which can be done by the induction hypothesis), we obtain a k-coloring of G’. Put m = w ( G ) , n = w ( H ) , and let p denote the maximum cardinality of a clique of G containing x. Then, obviously,

k =max{m,n+p-1). Consider an m-coloring of G and let K be the set of vertices having the same color as x. Let, further, L be a set of independent vertices of H meeting every n-element clique of H. Then T = L U ( K \{x}) is a stable set in G‘. Moreover, T intersects every k-element clique of G‘. Really, if C is a k-element clique of G’ and it meets H then, obviously, it contains an n-element clique of H and thus a vertex of L. On the other hand, if C does not meet H, then C must be an m-element clique of G, and thus C contains a vertex of K \ { x } . 0

As has been mentioned in the introduction, in view of Fulkerson’s results, the perfect graph theorem already follows from Theorem 1. However, to make the paper self-contained, we give a proof of the perfect graph conjecture (which seems to be different from that of Fulkerson).

2.

A hypergraph is a non-empty finite collection of non-empty finite sets called edges. The elements of edges are the vertices. Multiple edges are allowed, i.e., more (distinguished) edges may have the same set of vertices. The number of edges with the same vertices is called the multiplicity of them. The number of edges containing a given vertex is the degree of it. The maximum degree of vertices of a hypergraph H will be denoted by 8 ( H ) . A partial hypergraph of H is a hypergraph consisting of certain edges of H.

32

L. Loua'sz

The subhypergraph induced by a set X of vertices means the hypergraph H

[ x= { E n x 1 E EH, E nx~ra).

A partial subhypergraph is a subhypergraph of a partial hypergraph (or, equivalently, a partial hypergraph of a subhypergraph). The chromatic number x ( H ) of a hypergraph H is the least number of colors sufficient to color the vertices (so that every edge with more than one vertex has at least two vertices with different colors). The chromatic index q ( H ) of H is the least number of colors by which the edges can be colored so that edges with the same color are disjoint. Obviously,

4 (HI 2 8 (HI.

(4)

Let a hypergraph be called normal if the equality holds in (4) for every partial hypergraph of it. A set T of vertices of H is called a transversal if it meets every edge of H ; T ( H ) is the minimum cardinality of transversals. Denoting by v ( H ) the maximum number of pairwise disjoint edges of H, we obviously have U(H) T(H).

(5)

Let a hypergraph be called 7-normal if the equality holds in (5) for every partial hypergraph of it. A hypergraph is said to have the Helly property if any collection of edges whose intersection is empty contains two disjoint edges. It is easily seen that normal and T-normal hypergraphs have the Helly property. Given a hypergraph H, we can consider its edge-graph G ( H ) defined as follows: the vertices of G ( H )are the edges of H and two edges of H are joined iff they intersect. O n the other hand, for a given graph G we can construct a hypergraph H ( G ) by considering the maximal cliques of G (in the settheoretical sense) as vertices of H and, for any vertex x of G, the set of maximal cliques containing x, as an edge of H ( G ) . It is easily shown that if G has no multiple edges (which can be assumed throughout this paper) then G ( H ( G ) ) =G.

(6)

Furthermore, H ( G) always has the Helly property. It is easily seen that -

( G ( H )is the complement of G ( H ) ) .Moreover, if H has the Helly property then


33

Hence by (6),

X ( G )= s ( H ( G ) ) , w ( G )= s ( H ( G ) ) ,

x ( G )= T ( H ( G ) ) ,

4) = u(H(G))7

(9)

for any graph G. Equalities (7), ( 8 ) and (9) imply the following theorem:

Theorem 2. Let H be a hypergraph with the Helly property. H is normal iff G ( H ) is perfect; G is perfect if H ( G ) is normal. H i s r-normal iff G ( H )is perfect; is perfect i f H ( G ) is 7-normal.

As a corollary to Theorems 1 and 2 we have the following theorem: Theorem 1‘. Multiplying some edges of a normal hypergraph, the obtained hypergraph is normal. Theorem 2 implies that the perfect graph theorem is equivalent to the following:

Theorem 3. A hypergraph is r-normal iff it is normal. Proof. Parts “if” and “only if” of this theorem are equivalent (by Theorem 2). Thus it is enough to show that if H is normal then 7 ( H )= v(H)7 since for the partial hypergraphs this follows similarly. We use induction on n = r ( H ) . For n = O the statement can be considered to be true. It is enough to find a vertex x with the property that the partial hypergraph H’ consisting of the edges not containing x has v ( H ‘ )< u ( H ) ,since then H’ has an ( n - 1)-element transversal T and then T U {x} is an n-element transversal of H’, showing that

T ( H )n ~= v ( H ) . Assume indirectly that for any vertex x there is a system F, of n disjoint edges not covering x. Let

Ho=

U Fx7 x

where the edges occurring in more F,’s are taken with multiplicity. Ho arises from H by removing and multiplying edges, hence by Theorem 1’ it is also normal, i.e., q (Ho)= s (Ho).

L. Lovrisz

34

But obviously H,, has n . m edges, where m is the number of vertices of H. Since there are at most n disjoint edges in Ho, we have q (Ho)2 m.

On the other hand, a given vertex x is covered by at most one edge of F, ( y = x ) and by no edge of F,. Hence

S ( H o ) sm a contradiction.

- 1,

0

3. A subhypergraph of a normal hypergraph is not always normal as shown, e.g., by the hypergraph

{ { a ,b, d } , { b ,c, d } , I a ,c, 41; here { a , 6, c} spans a non-normal subhypergraph. Hypergraphs with the property that every subhypergraph of them is normal are described in the following theorem. A hypergraph is balanced if no odd circuit occurs among its partial hypergraphs (an odd circuit is a hypergraph isomorphic with the hypergraph ((1.2). { 2 , 3 } ,. . . ,(2n.2n + 1),11,2n + 1))).

Theorem 4. The following statements are equivalent : (i) H is balanced; (ii) every subhypergraph of H has chromatic number 2; (iii) every subhypergraph of H is normal. Obviously, Theorem 3 gives more equivalent formulations of (iiii). The theorem is actually due to Berge [3]. In what follows, we are going to give a new proof for the non-trivial parts of it.

Proof of Theorem 4. (iii) 3 (i) being trivial, it is enough to show (i) 3 (ii) and (ii) (iii). (I) Assume that H is balanced, though it has subhypergraphs which are not 2-chromatic. Let Hobe such a subhypergraph with minimum number of vertices. Consider the graph G consisting of the two-element edges of Ho: every vertex of H,, is considered to be a vertex of G. Now G is connected. Really, if V ( G) = X U Y, X f l Y = 0, X,Yf 0, and no edge of G joins a vertex of X to a vertex of Y, then considering a 2-coloration of Ho X and one of Hc, Y (by the minimality of Ho such 2-colorations exist) these

+

1

I


35

form together a 2-coloration of Ho,since every edge E of Ho with 1 E I > 1 has at least two points in one of X , Y, and then even in this part of it there are two vertices with different colors. Since H is balanced, G is obviously bipartite. Let G be colored by two colors. Since Ho cannot be colored by two colors, there is an edge E, with I E I > 1, of HI having only vertices of the same color. Let x, y E E, x # y. Since G is connected, there is a path P of G connecting x and y. We may assume that no further vertex of E belongs to P. Then the subhypergraph spanned by the vertices of P contains an odd circuit, a contradiction. (11) Now let H be a hypergraph with property (ii); we show it has property (iii). Obviously it is enough to show T ( H )= V ( H ) .

Let T ( H )= t and consider a minimal partial subhypergraph Ho of H with the property 7(Ho)= t. If we show that Ho consists of independent edges, we are ready. Suppose indirectly El : E2E Ho, x E El n E2. By the minimality of Ho, there is a ( t - 1)-element transversal T, of Ho\{E,},i = 1,2. Put Q = TIfl T,, R, = T , \ Q , S = R I U R 2 U { x } . Obviously, X E T , , hence I S I = 2 1 R , ( + l . Since Ho S is 2-chromatic by (ii), there are two disjoint subsets of S both meeting every edge E of Ho S with IE I > 1. One of them, say M, has at most [$I S I] = I R 11 elements. Now M U Q is a transversal of Ho. Indeed, if an edge E is not represented by Q then it meets both R 1and R, if E Z E, and meets RIP,and { x } if E = E, ; thus, I E n S 122, whence E is represented by M. But I M U Q I s I R I I + I Q I = t - l , a contradiction. 0

I

1

We conclude this section with the remark that bipartite graphs are, obviously, balanced (and thus normal). On the other hand, Theorem 4 shows that balanced hypergraphs have chromatic number 2. Recently, Las Vergnas and Fournier sharpened this statement and showed that normal hypergraphs have chromatic number 2.

4.

Let

be a (0, 1)-matrix, no row or column of which is the 0 vector, and consider the optimization programs

L. Lovdsz

36

yA 3 w v20 min y . 1 Ax 6 1 x 20

maxw.x where 1 denotes the vector

It is well-known that if x, y run through non-negative real vectors, (10) and (11) have a common optimum. But now we are interested in integer vector solutions. Let B be a (0, 1)-matrix such that (i) any column u of B satisfies Au C 1, (ii) every maximal (0, 1)-vector with this property is a column of B. Consider two further programs:

yB 3 w y 30 min y * 1

Bx s 1 x 20 rnaxw-x Theorem 5. Assume that the optimum of (10) ( = the optimum of (11)) is an integer for any (0, 1)-vector w. Then, for any non-negative integer vector w, each of (10)-(13) has an integer optimum and an integer solution vector. Remark. The greatest part of this theorem is formulated in Fulkerson [ 5 ] as a

consequence of the perfect graph conjecture and the theory of anti-blocking polyhedra. Proof of Theorem 5. (1) First we show that (11) has a solution vector with integral entries for any (0, 1)-vector wo. For let xo be a solution of it with the greatest possible number of 0’s. Put wn= ( w , , . . . , W k ) ,

xu = (XI) xk

Obviously, x:S

WO. We

show that xo is an integer vector.


37

Assume indirectly 0 < x 1 < 1, say; then w 1 = 1. Put

w: =

{

1 if x,#Oand i > l , 0 otherwise,

and

w ’ = ( w I, . . ., w ;). Let x’ be a solution of (11) with w

=

w’, then

w ’ x ’ s w o x ’ s woxo and

w ’x‘ 3 w ’xo > W”X0 - 1. Hence, both w’x’ and woxo being integers,

w‘x‘ = W(,X’ = woxo, i.e., x’ is a solution of (11) with w = wo too, and has, obviously, more 0’s than xo has, a contradiction. ( 2 ) Now we prove that also (10) has an integer solution vector for any (0, 1)-vector w. Assume indirectly that there are (0, 1)-vectors w failing to have this property and let wo be one with minimum number of 1’s. Let y o be a solution of (10) with w = wo. Obviously, we may assume that y l s 1. Put WO

= ( w l , ...9

Wk)r

Y O

= ( y l , . . . ? yk),

yl

# 0,

say, and define a (0, 1)-vector w’ = ( w I,. . . ,w ; ) by

w, if a l , = 0 , w: =

0

otherwise.

We show first that yo is not a solution of (lo), with w = w‘. For let x’ be a solution of (11) with w = w’; we may assume x I T S w’. Then

or yo(l - A X ’ = ) 0, but this is impossible since both yo and 1 - A x ’ are non-negative and their first entries are y l and 3 1 a,,w: = 1 , i.e., the inner product is non-zero. Thus, considering a solution y’ = (y I,. . . ,y:) of (10) with w = w‘ we have

-x:=l

38

L. Lovdsz

y'.l is said to be a pseudo-order on a set X if we have:

a > 6, b > c 3 a > c o r c > a (pseudo-transitivity), a>b 3 notb>a

(antisymmetry).

A graph G is a comparability graph if and only if it admits an orientation of its edges that represents a pseudo -order relation. This result yields a characterization in terms of odd closed walks which had been conjectured by A. Hoffmann:

Theorem 4.2 (Ghouila-Houri [34], Gilmore and Hoffmann [35]). A graph G is a comparability graph if and only if for every closed walk with odd length ~ X , exists an edge of the form x,x,+~(where i is taken modulo x I .* - X ~ ~ + there 2p + 1). Gallai strengthened this result, considering the following relation between adjacent edges : a b ac i f and only if b and c are in the same connected component of the subgraph GrGc,, induced by the neighbours of a (in G ) in the complement of the graph G. A wreath of G is defined to be a cycle x l . . x&l that satisfies the following two conditions: (i) all x, are different,

-

2K2

c.4

c5

D2

Fig. 6.

El

Classical perfect graphs

77

-

(ii) xixi-] xixi+lfor all i (modulo p ) , where p is the length of the wreath (see Fig. 7).

Theorem 4.3 (Gallai [32]). A graph G is a comparability graph if and only if it does not have a n odd wreath. We shall say nothing of the deep structure of comparability graphs for we would plagiarize Gallai’s remarkable work [32], the conclusion of which is the following characterization b y excluded configurations:

Theorem 4.4 (Gallai [32]). G is a comparability graph if and only i f it does not contain an induced subgraph isomorphic to one of the graphs C (1 d i d 4) shown in Fig. 8 ( a ) or of the complements of the graphs K (5 S i S 19) shown in Fig. 8(b). Recently, Comparability graphs were characterized as the complement of some intersection graphs:

Theorem 4.5 (Rotem and Urrutia [72]). The complement of a graph G with n vertices is a comparability graph if and only if G is the intersection graph of the graphs f , , . . . ,f n of n continuous functions E : (0,l)- R. In fact, Rotem and Urrutia proved that the E’s may be chosen linear by intervals and intersecting in a finite number of points. Arditti et al. [2] studied the comparability graphs that completely determine (up to duality) a partial order. These graphs are named uniquely partially orderable graphs (U.P.O.). Such graphs have been characterized by Shevrin and Filippov [74] and Trotter, Moore and Sumner [78]. Theorem 4.6 (Arditti [2]). The comparability graph of an irreducible partial order is U.P.O.

Fig. 7. Wreaths.

P. Duck1

78

2n

2n 0 2 2ntl 1 2n+ &$2 1

1

"L/ n

1

D,

C, n > 6

n>2

En n > 1

F, n > 1

,/A A1

A3

A4

A5

dz A7

A0

A9

A10

Fig. 8(b). Fig. 8. Critical non-comparability graphs @(a)) or their complements (U(b)).

More generally, the comparability graph of a partially ordered set suffices to give much interesting information about this order. Theorem 4.7 (Trotter, Moore and Sumner [78]). The dimension of a partial order ( = the minimum number of linear orders whose intersection is the given order) is determined by its comparability graph.

Theorem 4.7 also appears in [47]. A nice representation of a comparability graph, generalizing the result on permutation graphs by Dushnik and Miller (see Theorem 3.3), is given by the following theorem:

Clnssicnl perfect graphs

79

Theorem 4.8 (Leclerc [57]). G is the comparability graph of an order of dimension < d if and only if it represents the inclusion relation between subtrees of some tree that has at most d pendant vertices. (In this theorem, a ‘subtree’ means a subset of vertices that induces a subtree; see Section 5.) Purely graphical strong properties, generalizing perfectness, were obtained by Greene and Kleitman [38], [40]. See [39] for a survey of these properties involving cliques and stable sets (or, equivalently, chains and antichains for partial orders); these results generalize to a great extent the famous Dilworth theorem (see [13], [28], [64], [65], [76], [30],[41], [SO]). Using the minimal cost flow algorithm of Ford and Fulkerson [25],Frank [28] gives a nice formulation and a generalization of Greene and Kleitman’s min-max relations. Following Frank, let us say that two families % and Y constituted by subsets of a given set X are orthogdnal when they satisfy the following two conditions:

C nS#

0

for every C E % and every S E 9,

(4.10)

Theorem 4.11 (Frank [28]). There exist families V1,. . . , Va and Y 1 ,... ,Yo and there exist integers a = j , > * * > js = 1 and 1 = i, < * < is = w with the following properties : i s a family of j disjoint cliques of G (for 1 S j d a ) . (i) (ii) Y, is a family of i disjoint stable sets of G (for 1 S i S w ) . and Y,, are orthogonal (1 k S s). (iii) For every j, jk 2 j 2 jk+,, (iv) For every i, ik s i s i k + l , Y, and V1,are orthogonal (1 G k s s). Corollary 4.12 (Greene and Kleitman [40]). Denote

where the maximum is taken over all families Y, consisting of i stable sets of G. Then

where the ,first minimum runs over all partitions of X into cliques C1,.. . , C,. Corollary 4.13 (Greene [38]). Denote

P. Duchet

80 w,

=max el

1 uc/, C€’%,

where the maximum is taken over all families %, consisting of j cliques of G. Then w,

=min Cmin(lS,I,j) t=1

where the first minimum runs over all partitions of X into stable sets S,, . . . ,S,.

5. Representation of triangulated graphs A subtree means here a subset of vertices that induces a subtree of a given graph. A subtree hypergraph is a finite hypergraph H = (X, %) whose edges are subtrees of some fixed tree having X as set of vertices.

Theorem 5.1 (Duchet [ 161, Flament [23]). A hypergraph H = ( X ,%) is a subtree hypergraph if and only if it has both of the following properties: (S.l.1) H is u Helly hypergraph. (5.1.2) Every cycle of H contains three edges with a non-empty intersection. It should be noted that a subtree hypergraph is not generally a balanced hypergraph.

Proof of Theorem 5.1. The proof of this theorem uses the following simple remark whose verification is left to the reader:

Lemma 5.2. Let E and F be two intersecting edges of a hypergraph that satisfy (5.1.1) and (5.1.2). Then, if we add E U F o r E f l F as a new edge, the resulting hypergraph also satisfies (5.1.1) and (5.1.2). Now, among all hypergraphs containing H as a partial hypergraph and satisfying conditions (5.1.1) and (S.1.2), we choose one, say H‘, with a maximal number of edges. Let T be the partial hypergraph of H’ constituted by the minimal edges that contain at least two vertices (‘minimal’ is relative to the inclusion relation). We are going to show that T is the required tree; H’ is a family of subtrees of T. Let E be an edge of H’, IE 122, and a, b two vertices of E. Suppose { u. b ) E H ‘ . Then, by maximality of H’, if we add { a ,b } as a new edge of H’, the resulting hypergraph contradicts (S.l.1) or (S.1.2). In both cases, noting that if


81

a E l x l* . . x k +Ekb , is a path in H‘ from a to b, then E , U * * * U E, is an edge of H’ by Lemma 5.2, there exist two edges of H’, say A and B,with the properties a€A, bEB

and

AflBn{a,b}=0.

By (5.1.1) we have A n B n E # 0, hence A n E and B f l E are edges of H’ with at least two vertices each. We may conclude as follows: ( I ) If E is in T, then E = { a , b } . Hence, T is a graph and, by (5.1.2), has no cycle. (2) If E’ is every edge in H ’ , every two vertices a and b of E‘ are connected by a path in H’\E‘ whose edges are included in E ’ . The conclusion easily follows, by induction on E’l . 0

I

A complete description of all trees on which a subtree-hypergraph is representable is possible (see [16], [17]). The above theorem provides a nice characterization of triangulated graphs, many times rediscovered: Theorem 5.3 (Buneman [ 101, Gavril [33]). A graph G is a triangulated graph if and only if G is a subtree graph, i.e., the intersection graph of a family of subtrees of a tree. Moreover (see [lo]), a triangulated graph is the intersection graph of subtrees of a tree whose vertices are the maximal cliques of G, the subtrees being constituted by the set of those maximal cliques that contains a given vertex of G. The proof is a simple application of Lemma 2.3 and Theorem 5.2, noting that G is a triangulated graph if and only if C*(G)satisfies (5.1.1) and (5.1.2). An example of representation by subtrees is shown in Fig. 9. Walter [81], [82] studied some graphs that are intersection graphs of subtrees of a prescribed tree.

6 Fig. 9.

4

5

a2

P. Duchet

Gavril[33] remarked that every triangulated graph is the intersection graph of a 'Sperner' family of subtrees of a tree ( = no subtree is included in another).

Corollary 5.4. A hypergraph H is the set of all maximal cliques of a triangulated graph if and only if it satisfies one of the following equivalent properties: (5.4.1) No edge of H contains another edge of H and in every cycle of length 3 3 of H there is an edge that contains three vertices of the cycle (Berge, cited in I1 1). (5.4.2) No edge of H contains another edge of H and the dual hypergraph H * is a subtree hypergraph. Analysing a cyclomatic number for hypergraphs, Acharya and Las Vergnas [ 11 have shown another interesting characterization of these hypergraphs. More precisely, let H be a hypergraph: L , ( H ) is the graph L ( H ) with weighted edges (the weight of (E, F) being I E n F I ) ; w ( H ) denotes the maximal weight of a forest of L , ( H ) ; and p ( H ) , the cyclomatic number, is defined by

(5.5) Theorem 5.6 (Acharya and Las Vergnas [I]). The cyclomatic number of a hypergraph H equals zero if and only if its edges, maximal with respect to inclusion. are the maximal cliques of a triangulated graph. Theorem 5.7 (Duchet [16]). Euery triangulated graph is representative of a family of convex polygons in the plane. Proof. Choose a tree T with vertex set X. T admits a planar embedding such that the points corresponding to vertices of T are exactly the vertices of a convex polygon. Denote by f : X + R Z t h e vertex embedding. The reader can easily verify that, for every two subtrees A and B of T, we have A f l B # 0 if and only if Conv ( f ( A)) f l Conv ( f (B)) # 0,where Conv denotes the ordinary convex hull. The conclusion follows by Theorem 5.3. 0 An interesting problem o n representative graphs of subtrees has not yet been completely solved:

Theorem 5.8 (Renz [67]). G is the intersection graph of a family of paths in a tree if and only if G is triangulated and is the intersection graph of a family F of paths of a graph such that F satisfies the Helly property.


83

6. Graphical properties of triangulated graphs

Various properties of triangulated graphs have been proposed. The most important one was discovered by Dirac [14] who first proved the existence of a simplicia1 vertex. This fact is at the origin of an elimination scheme in solving linear systems (see Rose [70] and Golumbic [36]). Let us give a few basic definitions: Definitions. Asimplicial vertex of a graph G is a vertex all of whose neighbours are adjacent. A minimal relative curser (relative to x and y ) is a subset of vertices whose suppression disconnects two fixed vertices of the graph ( x and y ) and which is minimal for the inclusion order with this property. In Fig. 10, { a , b } is a minimal relative cutset. A T-orientation (called ‘monotone transitive orientation’ in Rose [70]) of a graph G is an orientation of the edges of G such that the following two conditions are fulfilled: (1) The orientation has no circuit. implies

b-c

b +.a.

A path a = x , . . . x,, = b from a to b is named a minimal path when it has no chords; no edge in the graph is of the form xixj with i + 1 # j. Three technical, but simple lemmas will be useful. In what follows, G denotes a triangulated graph. Lemma 6.1. Every closed walk x 1 * * . x, . . . x,xI with n x , x , + ~or a repetition x , = x , + ~ .

Fig. 10.

32

= ( X ,E

)

has a triangular chord

84

P. Ducher

Lemma 6.2. Let a and b be two neighbours of a vertex x in G. lf x i = a . . . x, . . . x,, = b is a minimal path not passing through x, then all x, are neighbours of x.

Lemma 6.3. Suppose V and C are disjoint subsets of vertices in G. Suppose V is connected and C is a maximal clique. Then C contains a vertex not adjacent to V. Proofs. Lemma 6.1 and 6.2 are easy to verify and left to the reader. To prove Lemma 6.3, choose in V a subset W that is maximal with the properties that W is connected. and C contains a vertex s not adjacent to W. If W # V, consider in V \ W a vertex u having a neighbour w in W. C contains a vertex t not adjacent to t‘. Clearly, t is distinct from s and has a neighbour w’ in V. Moreover, s is adjacent to u. A minimal path uw . * w , . * . w‘t, where the w,’s are in W , contradicts Lemma 6.2. 0

Theorem 6.4. Each of the following assertions is a necessary and sufficient condition for a graph G = ( X , E ) to be triangulated: (1) (Dirac [14]) Every minimal relative cutset of G is a clique. (2) (Frank-Kas [29]) Every induced subgraph of G is either a clique or contains two non -adjacent simplicia1 vertices. ( 3 ) (Dirac [ 141, Lekkerkerker and Boland [%I) Every induced subgraph has a simplicia1 vertex. (4) (Rose [70]) There exists a n order x , , . . . ,x, of vertices of G such that every x, is a simplicia1 vertex of the induced subgraph G, , x m , for i = 1 , . . .,n. ( 5 ) (Rose [70]) G admits a T-orientation of its edges. (6) (Duchet [ 151) Every connected subgraph with p 2 2 vertices contains at most p - 1 cliques. The formulation of (1) makes precise a result of Hajnal and Surangi [48], who have shown a weaker property of triangulated graphs (sufficient to prove the perfectness): every minimal cutset is a clique.

Proof of Theorem 6.4. The proofs of (1) and ( I ) 3 (2) are simplified versions of cited proofs. The implication (5) j (6) is new. Let x, y be two non-adjacent vertices of a triangulated graph G, S a minimal cutset relative to x and y. and a, b two non-adjacent vertices of S. If C, and C, denote the connected components of Gx-s containing, respectively, x and y. there exists, by minimality of S, a minimal path from a to b in C, U { a ,b } and a minimal path from a to b in C, U { a , b } . These paths form a cycle of length 2 4 with no chord. This contradiction proves (1).


85

(1) j(2) is shown by induction on the number of vertices. If x and y are non-adjacent vertices of G, if S is a minimal cutset relative to x and y , and if C, and C, are the associated connected components, then the induced subgraph GcXushas two simplicial vertices that are not adjacent and one of them is a simplicial vertex of G. The same argument holds for Gcyus. (2) j(3) 3 (4) j ( 5 ) are trivial. (5)j (6) does not present any complication. Let G denote a T-orientation of G. The sets C, = { x } U TC(x)are cliques of G. Conversely, a clique of G induces a transitive tournament in G and so is contained in a set of the form C,. Thus, all the maximal cliques in G are of the form C,. A source in G does not produce a maximal clique. The conclusion follows. (6) implies trivially that every cycle of G, with length 3 4, has a chord. 0

Remark. The ratio k ( G ) / a ( G )where , k ( G )is the number of maximal cliques and a ( G ) is the stability number, is not bounded for triangulated graphs, as shown by the graph Gn,"having 2n vertices a l,..., a,,, b , , . .. , b,, and the following edges: a,a, for all i, j , a,b, for i < j , 6,b, for all i, j. For this graph, k(G,.,)= n and a ( G )= 2. In the following corollaries, G

= ( X ,E

) is a triangulated graph.

Corollary 6.5 (Lekkerkerker and Boland [58]). For every subset A of X , the subset X \ A \ r G ( A ) is either empty or contains a simplicial vertex of G. Proof. The proof is easy by 6.4(1) and 6.4(2). 0 Corollary 6.6 (Laskar and Shier [56]). For every vertex x of G there exists a simplicial vertex y at maximum distance :

d ( x , y ) = max d ( x , z ) . 1€X

Proof. The proof is by induction on the maximum distance, using the above corollary. 0

Various properties involving distances and centers of triangulated graphs have

P. Ducher

86

been investigated by Laskar and Shier [56]. They conjectured that every odd power of a triangulated graph is triangulated. In fact, we prove a slightly stronger result. In the following theorem, G k denotes the k-th power of G, that is, the graph with the same vertices as G, two vertices being adjacent in G' when their distance in G is at most k. Theorem 6.7 (Duchet). Let G be a graph. If G' is triangulated, so is G k + 2 .

Lemma 6.8. If G = ( V , E ) is a triangulated graph, and if VI,. . ., V, are connected subsets of V, G ( VI,. . . , Vm)is also a triangulated graph. Proof of Lemma 6.8. Let c = Cl . . . ckcl be a cycle with length 3 4 in G ( V,, .- ., V , ) . A closed walk uI * * up of G is called a C-walk if and only if uI"'up=

WI &

wz &

... &

Wk

& WI

(& denotes the concatenation of sequences) where W, is a non-empty walk in the subgraph induced by G over C,. W, is called the i-th component of the C-decomposition W , ,. . . , W,. Consider a C-walk W with minimal length p . Denote by u I , . . . ,v,, the vertices of W and W , ,. . . , wk the C-decomposition of W. By Lemma 6.1, W must contain two vertices u,, and uqr2 such that

u ~ ,= uqA2 or

u, is adjacent to uq+?.

The minimality of W implies that u, and uq+2are in different components W, and W, with

Ia-PI#l,

(wP)#(k,l)

and

(a,P)#(l,k).

Therefore W., and W, are linked in G ( V l , .. ., V , ) and the lemma is proved. 0 Proof of Theorem 6.7. f U I . . . . ,u, are vertices of G, Gk" is clearly isomorphic to G k ( V,..., l V m )where Vi = r G ( v i ) U { v i } So, . we are done. 0 Definition. A block graph is a graph whose blocks are cliques.

Corollary 6.9 (Jamison [53]). If B is a block graph, B k is triangulated for all k 3 1. Corollary 6.10. If G is a triangulated graph, so is G"" for euery integer k. The property fails for even powers (see [56]).


87

7. Representations by intervals

An interval hypergraph is a hypergraph isomorphic to a (finite) family of intervals of N. In other words, there exists a linear order o n the vertices of H such that every edge of H is an interval for this order. Algorithmic characterizations have been proposed by Fulkerson and Gross [31], Eswaran [20], and others. Theoretical characterizations are due to Tucker [79], Duchet [15], [ 161, Fournier [27], and Nebesky 1621. The following form seems to be most pleasant: In a hypergraph, a vertex a is said to lie between vertices b and c when every path joining b to c contains an edge containing a. Theorem 7.1 (Duchet [15]). A hypergraph H is an interval hypergraph i f and only if the following condition is fulfilled:

For every three vertices H, one of them lies between the other two.

(1)

Proof. The proof is similar to that of Theorem 5.1 and is much simpler than in 1271, 1151, [621. The following lemma is useful. Lemma 7.2. Let E, F be two intersecting edges of a hypergraph H that satis,fies ( I ) . Then, i f we add E U For E f l F as a new edge of H, the resulting hypergraph also satis,fies ( I ) . If E,C F and F Z E, a similar property holds with E \ F. The verification of this lemma is straightforward and left to the reader. Proof of Theorem 7.1 (continued). Now let H satisfy (I). Among all hypergraphs that satisfy (I) and that contain H as a partial hypergraph, we choose one, say H’, with a maximal number of edges. Let T be the hypergraph whose edges are the minimal edges of H’ with at least two vertices. We are going to show that T is a path and that H is a family of subintervals of this path. Let E be an edge of H’, I E 122, and let a, b be two vertices of E. If E # { a , b } , the maximality of H ‘ implies the existence of three vertices x, y, z and three paths xEl * . . E,y, yFI . * Fpz, zG1* . . G,x satisfying

ZP

Ifi E,,

,=I

XE

,=I

F,, z E

u G,.

k=i

{ a , b } is one of these edges. Moreover, we may suppose the following:

P. Duchet

88

(*) Property (I) holds when any occurrence of { a , b } in the sequence E, . . . F, . . Gk is replaced by every edge of H' that contains both a and b. Thus, changing { a , b ) to E, we have, for instance, that z lies between x and y . Hence a, h. z are different, { a , b } is some F, and z E E. By Lemma 7.2, there are edges A and B of H' such that

a E A, z P A, and A contains, for instance, x. b E B,

Z PB,

and B contains

Y-

Moreover, b e A, otherwise A may replace E, = { a , 6) in t h e path joining x and y, in contradiction to our assumption (*). Similarly a @ B. Thus E \ B and E \ A are edges of H' connecting a and b. We may therefore conclude with the following statements: (1) If E is an edge of T, E = { a , b } . Hence, T is a graph. (2) Every pair of vertices of an edge E in H' are linked by a path in H' whose cdges are included in E. X being an edge of H'. T is connected and, by (I) applied to T, is a path. By induction on I E I , every edge of H' is a connected subset of T. 0

Theorem 7.3 (Tucker [79]). His an interval hypergraph if and only if it does not contain as induced subhypergraph one of the list in Fig. 11. Tucker's proof is long. Using the characterization of interval graphs by Lekkerkerker and Boland [%I, whose proof is also long, Trotter and Moore [77] gave a short proof. We give here a direct short proof as a consequence of the above characterization.

c,

n r 3

01

N,

M, n > 1

n> 1

Fig. 11.

Chsical perfect graphs

89

Proof of Theorem 7.3. The hypergraphs of Fig. 10 are clearly not interval hypergraphs. Conversely, suppose H = ( X , ‘8‘) is not an interval hypergraph, every proper subhypergraph of H being an interval hypergraph. Let x be a vertex of H. Since H does not satisfy (I), there exist two vertices y and z in X \ x and three paths xEl . . . E,y, xF, . . . F,z, y G 1 .. . G,z with the properties

XP u k=l

G,,

ye 6 F,, /=I

ZE

6 E,.

,=I

We choose p and q as short as possible over y, z and all such systems of paths (x remaining fixed). If El n Fl = (?,H contains some cycle C,,. If El C F, (the case F, C El follows by symmetry), we easily find 01,O2or N ,

as induced subhypergraph. If El f l F1# 13, FIf El and E l ( F1# (?, the minimalities of H, p and q imply y E E, \ FI and z E FI\ El. The subhypergraph induced by H over the set Gk U (El f l F , ) contains C, or M,, as partial subhypergraph. 0

uL=,

An interesting property of families of intervals was discovered by Fulkerson and Gross [31], namely, that an interval hypergraph is completely determined by its ‘intersection pattern’. Theorem 7.4 (Fulkerson and Gross [31]). Let (E,),E,be a ,finite family of intervals of N, and let (E),El be a family of sets. Suppose J E ,n El 1 = ( E f l F, 1 forall i, j E I. Then the families E, and E are isomorphic.

(7.5)

This theorem was considerably strengthened to more general hypergraphs by Fournier [26]. Nevertheless, the problem of characterizing the matrices 1 E, f l El I of families of intervals is still open. Interval hypergraphs are unimodular (Berge [6]). A nice min-max relation for intervals was recently settled by Gyori [46]. Other kinds of representation via intervals have been proposed, such as ‘D-interval hypergraphs’ by Moore [61], ‘dotted-intervals’ by Duchet [ 151, and ‘interval representability’ by Trotter and Moore [77]. See also Gyarfis and Lehel [451. 8. Interval and indifference graphs

Hajos [49] was the first to note the deep interest of ‘intersection graphs of families of intervals of the real line’. Hajos’ graphs became interval graphs and

!MI

P. Ducher

play a part in various applied fields like psychology (distinguishability in hierarchies), zoology (place of species in animal evolution), genetics (gene as segment of a chromosome), archeology (seriations), geology (classifications), criminology (see [9]), etc., in fact, wherever a chronological ordering is deduced from data which only indicates contemporaries. See [3], [42], [43], [44]. [60], [4], [54], [69], [7S]. Interval graphs have many interesting properties. They are unirnodular graphs (see [S]), triangulated graphs (Hajos [49]), and their complements are comparability graphs [49]. A first characterization may be formulated in terms of betweenness, as follows. In a graph G, a vertex x is said to lie between the vertices y and z when every path joining y to z must contain x or a neighbour of x.

Theorem 8.1 (Lekkerkerker and Boland [%I). A graph G is an interval graph if and only if it satisfies the following two conditions: (i) G is n triangulated graph. (ii) Among every three vertices of G, at least one is situated between the others. Moreover, condition (ii) may be replaced by the following weaker form: (ii') Among every three simplicia1 vertices of G at least one is situated between the others. Theorem 8.2 (Lekkerkerker and Boland [%I). A necessary and sufficient condition for a graph G to be a n interval graph is: (iii) G does not have as induced subgraph one of the graphs listed in Fig. 12. Proofs of Theorems 8.1 and 8.2. The class of interval hypergraphs is clearly an R-class (see Section 2) and Lemma 2.3 can be applied. Conditions (i) and (ii) are clearly necessary for a graph to be an interval graph. Conditions (i) and (ii') together obviously imply (iii). For proving the sufficiency of (iii), suppose G is a triangulated graph but not an interval graph. Then, by Theorem 7.3, the hypergraph C*(G ) possesses as induced subhypergraph one of the hypergraphs O , , O?, M,, N,, C,, (see Fig. 11).

Fig. 12.

91


Thus, applying Lemma 6.3, we have the following: If C*(G) contains 01,G contains A 2 or D2. If C*G contains 02,G contains A 1 or Dz or D3. If C*G contains M,, G contains some D,. If C*G contains N., G contains some E,. If C*G contains C,,, G contains E , or Ez. 0 Corollary 8.3 (Gilmore and Hoffman [35]). G is an interval graph if and only if G is a triangulated graph and the complementary graph is a comparability graph. Proof. None of the graphs A l , A2, 0, and Ep is a comparability graph. This proves the ‘if’ part. It is not difficult to verify that the conditions are necessary. If G = I!,(%‘), where 8 is a family of intervals in N, put I < J for two members I and J of 8 when

InJZ0, (VX E I ) ( V y E J ) x < y. The comparability graph associated with the partial order < is

G. 0

Such orders are termed interval orders and have been characterized by Fishburn [22] as partial orders < that satisfy

(8.4) ‘‘x < y and z < t” implies ‘‘x d t or z < y for every element x, y , z, t. ”

Corollary 8.5 (Duchet [15]). G is a n interval graph if and only if it has no induced C, and if for every closed walk x I * x,xl with n 3 5 there exist partitions A = {xPcl. . . x q } , B = {x,+, * * x,} (the indices are modulo n ) that satisfy both following conditions : (1) Every vertex of A has a neighbour in €3. (2) Every uertex of B has a neighbour in A. Corollary 8.6 (Kotzig [55]). The connected bipartite interval graphs are exactly the caterpillars ( = trees that become paths by removing the pendant vertices).

The proofs are left as exercises. We cannot conclude this section without mentioning an important family of interval graphs, the indifference graphs. Definition. Let 9 be any family of intervals of the real line and X be a finite

P. Duchet

92

subset o f points of the line. Link x and y by an edge if some member of 5!? contains both x and y ("indistingiiishability"). This forms a graph on %. The indifference graphs are all the graphs that can be obtained in this way. An equivalent definition (using the tcrminology of Berge [7]) is the 2-section of an interval hypergraph. In the context of Decision Science, Luce [60] introduced the notion of a semi-order.

Definition. A partial order < on a set X is called a semi-order if there exist a real number d > 0 and a real-valued function n : X - R with the following property: x < y e Lf(X)B u(y)+d. A necessary and sufficient condition for < to be a semi-order is (Scott and Suppes 1731) that < satisfies both condition (8.4) and the following condition: (8.7) "x

i y

and y < z"

+ "x < t or t
3,

Fig. 2.

100

C. Grinstead

neither a a , nor a3,zis adjacent to either u12or a1.3. Also, a2,1is not adjacent to a3.2,and al.2is not adjacent to a23. Since a2.2must be in four cliques of size four and no cliques of size five, H must contain four triangles and no cliques of size four. It is easy to check that this implies that every vertex in H must be in at least one triangle. This implies that a2I is adjacent to a1.3, for otherwise a2.1would be in no triangle in H. This implies that s = 2, which contradicts Theorem 4. If r = 3, then rs = I G I = 4 w ( G ) + 1; so s must be odd, and since we may assume that w ( G )2 4, from Theorem 3, then we may also assume that s 3 7. But then the vertices al.,,for 1S i S s, form an odd chordless cycle, which contradicts the fact that G is critical. Since I G I is odd, r # 2, so we may assume that r = 1. We may now drop the first subscript on all of the vertices. We have vertices al, az,. . . , us,where a, is adjacent to a, if i - j E D = { k 1, t, 2 ( t + l)},where all arithmetic is modulo s. From Theorem 4, we know that f f 0, - 1 (mod s), and since G is regular of degree six, t f 1 (mod s). We may also assume without loss that 1 < t < s2- 1. If f > 2, then a, is not adjacent to a, or as-l.Since a lmust be in a clique of size four with a,, either al is adjacent to us-,-l or us-, is adjacent to a,+l.Figure 3 shows the neighbors of a,. So, either s - t - 2 or s - 2t - 1 is in D. If s - f - 2 is in D,then it equals either t or t + 1. However, since s is odd, we must have s - t - 2 = t + 1, or s = 2t + 3. In this case, we can multiply all subscripts by 2, since ( 2 , s ) = 1, and get that a, and a, are adjacent if and only if i - j E D‘ = { & 1, k 2, ? 3). We will deal with this case at the end of this proof. If s - 2t - 1 is in D, then either s - 2t - 1 equals 1, t, or t + 1.It cannot equal 1, for s is odd. If it equals t, then s = 3t + 1. In this case, we can multiply all subscripts by 3, since (3, s ) = 1, and get that a, and a, are adjacent if and only if i - 1 E D’. This is the same conclusion as before. If s - 2t - 1= t + 1, then s = 3 t + 2 , so we can multiply all subscripts by 3, and arrive at the same conclusion as before. If t = 2, we get the same conclusion again. So, we are left with the case that G has s vertices and that a,and a, are adjacent if and only if i - j ED’. We may

*

Fig. 3.

The Perfect Graph Conjecture for toroidal graphs

101

assume that s 17, since o(G) 4, o(c)2 4. In this case, it is easy to find an odd chordless cycle. If s = 4m + 1, then the cycle is given by the vertices a,, a3, a+ a,, as, , . . , a4m-,a4m-2,a,. This contradicts the fact that G is critical and completes the proof. 0

References [l] A. Altshuler, Construction and enumeration of regular maps on the torus, Discrete Math. 4 (1973) 201-217. (21 C. Grinstead, The strong perfect graph conjecture for a class of graphs, Ph.D. Thesis, UCLA, Los Angeles (1978). [3] L. LovSsz, A characterization of perfect graphs, J. Comb. Theory, Ser. B 13 (1972) 95-98. [4] M. Padberg, Perfect zero-one matrices, Math. Program. 6 (1974) 180-196. [5] A. Tucker, The strong perfect graph conjecture for planar graphs, Canad. J. Math. 25 (1973) 103-114. [6] A. Tucker, Critical perfect graphs and perfect 3-chromatic graphs, J. Comb. Theory, Ser. B 23 (1977) 143-149.


Annals of Discrete Mathematics 21 (1984) 103-113 @ Elsevier Science Publishers B.V

THE PERFECT GRAPH CONJECTURE ON SPECIAL GRAPHS - A SURVEY* Wen-Lian HSU Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, Illinois, USA We discuss several basic techniques for proving the strong perfect graph conjecture o n special classes of graphs. Our discussion is primarily based on the neighborhood structure of these graphs. By combining these techniques we are able to prove the conjecture for more general graphs. The main classes of graphs we consider are claw-free graphs, 3-chromatic graphs, and (K,-e)-free graphs.

1. Introduction

In this paper all graphs considered are simple, i.e., finite, undirected, without loops or multiple edges. Let graph G = (V, E ) have respective vertex and edge sets V and E. Let a (G) denote the maximum size of a stable set of G and w ( G ) denote the maximum size of a clique of G. Let 8(G) denote the minimum number of cliques which cover G, and y ( G ) denote the minimum number of stable sets which cover G ( y ( G )is also called the chromatic number of G ) .Since two vertices of a clique cannot be in the same stable set, we have w ( G ) s y ( G ) . Similarly we have (Y (G) < 8 (G). A graph G is perfect if w ( H )= 8 ( H )for every induced subgraph H of G. An odd hole is a cordless odd cycle, and an odd anti-hole is a complementary graph of an odd hole. Berge’s (see [l])strong perfect graph conjecture (SPGC) asserts that a graph G is perfect if and only if G contains no odd holes or odd anti-holes. Its resolution has eluded researchers for two decades. One line of attack on this conjecture has been to look at general properties of critically imperfect graphs (p-critical graphs) - graphs that are not perfect but all of whose vertex-induced subgraphs are perfect. The major results here are as follows:

Theorem 1 [9]. A graph G with n vertices is perfect if and only if a(H)w(H)s 1 H 1 for all subgraphs H of G, where 1 H I is the number of vertices in the graph H. Thus a critically imperfect graph has n = CY ( G ) w ( G )+ 1 vertices. * This research was supported by National Science Foundation Grant ECS-8105989 to Northwestern University. in3

104

W.-L.HSU

Theorem 2 [lo]. (Let a = a ( G ) ,w = w ( G ) . ) A p-critical graph G with n vertices has exactly n cliques of size w with each vertex in o maximum cliques and has exactly n stable sets of size a with each vertex in a maximum stable sets. Each maximum clique intersects all but one maximum stable sets, and vice versa. Theorem 3 [2]. For each vertex u of a p-critical graph G, there is a unique partition of G - v into a-stable sets. In particular, the induced subgraph on any two a-stable sets in G - v must be connected. An example discovered independently by Huang [8] and by Chvital et al. [3] shows that the above three theorems alone are not sufficient to prove the SPGC (note that the addition of Proposition 1 to the properties of p-critical graphs will eliminate the counterexample). Therefore further characterization of p-critical graphs are desired. Another approach has been to look for special graphs which warrant a proof of the conjecture. In this paper we will combine several proof-techniques for special graphs and apply them to more general graphs with ‘good’ neighborhood structure. We will also discuss some relationships among various special graphs.

2. Some interesting neighborhood structures

Consider a graph G and a vertex u in G. Let N ( u ) be the set of vertices that are adjacent to u. For convenience, - we also regard N ( u ) as the induced subgraph on the neighbors of u and N ( u ) , its complement. Consider the following four neighborhood structures: (i) N ( u ) is bipartite, in which case u is called a b-vertex. (ii) N(u) is bipartite, in which case u is called a i-vertex. (iii) __ N ( u ) is complete multipartite, in which case u is called a m-vertex. (iv) N ( u ) is complete multipartite, in which case u is called a m-vertex. It is clear that these structures are inherent upon induced subgraphs. Corollary 1 implies that if every vertex of G is one of these four types, then the SPCC is true for G. We will prove several propositions and lemmas which together give Corollary 1. The following lemma is needed in our later proofs.

Lemma 1. A n odd cycle without triangle must contain an odd hole. Proposition 1. I f a p-critical graph G contains a b-vertex u, then G is an odd hole or an odd anti-hole.

The Perfect Graph Conjecture on special graphs

105

Proof. Since N ( u ) is bipartite, a maximum clique containing u has size 9 3 . Hence w ( G ) S 3 and the result follows from Tucker [15]. 0 Proposition 2. If a p-critical graph G contains a m -vertex u, then G is an odd hole or an odd anti-hole. Proof. Suppose N ( u ) is complete K-partite, then a maximum clique containing u has size K + 1. Hence o ( G ) = K + 1 and y ( N ( u ) )= K + 1. There must exist a part of N ( u ) which uses two or more colors. Consider two colors PI, p2 that are used in the same part C,. Since N ( u )is complete multipartite, no other part can use p , or p2. Let GPla be the subgraph of G \ u induced on vertices colored p i and p2.Since G is p-critical by Theorem 3, Galamust be connected. Let P be a shortest (PI,&)-path between two vertices x, y in N ( u ) with colors of x and y being p1and p2 respectively. Since x and y are in C,,they are not adjacent. Since P is the shortest possible such path, no other vertices in P are in N ( u ) . Therefore we have an odd hole uPu in G. 0 A claw is a bipartite graph Kl,3with four vertices vo, v I , u2, v 3 (Fig. 1)such that

the only edges that exist among them are (vO,u,), (vu,vz), ( u o , ~ s ) We . use {v0, ( v l , uz, v3)} to denote a claw rooted at uO.

Fig. 1. A claw

Proposition 3. Consider a p-critical graph G. If every vertex of G is a 6-vertex, then G is an odd hole or an odd anti-hole. ~

~

Proof. For each vertex u in G, N ( u ) is bipartite. Hence N ( u ) does not contain triangles. Thus in G, there can be no claw (induced subgraph) rooted at u. Therefore G is claw-free and the result follows from [12]. 0

Before stating a proposition about the m-vertices, we define the following two types of vertices. Type I. A vertex vo is a Type I vertex if there exist three other vertices 01, u2, ( u t , %), (u3, u*) E E and ( u l , u2)$Z . E (as shown in Fig. 2). It is easy to check that such a vertex is a m-vertex. u3 with (uo, uJ, ( U O , U Z ) , (uo, v3),

106

W.L. HSU

Fig. 2. Forbidden subgraph containing uw

Type 11. A vertex vOis a Type I1 vertex if there does not exist three other vertices u I , v2, v3 with ( V O , v,), ( V O , V Z ) , (v3, vl), (v3, V Z ) , (vt, ~ 2 E) E and (oo, v 3 ) eE (as shown in Fig. 3). We call u a m’-vertex.

Fig. 3. Forbidden subgraph containing uo.

It is clear that if every vertex of G is either a m”-vertexor a fi-vertex, then G is K4\e (K4 taking out one edge) free. Hence we can derive the following proposition from [ 1 I].

Proposition 4. Let G be a p-critical graph. If every vertex of G is either a m-vertex or a mi’-vertex, then G is an odd hole or an odd anti-hole. The distinction between m ‘-vertices and *-vertices will be important when we consider more general graphs in the next section. Previous SPGC proofs on special graphs rely heavily on the uniformity of those special properties (which often follow ChvBtal’s approach [4]) and, hence, are difficult to be used for a general proof of the SPGC. Our method is based primarily on a simplified version of an alternative proof (see [7]) that the SPGC is true for claw-free graphs, which uses the neighborhood approach.

3. Special graphs with mixed neighborhood properties

Theorem 4. Let G be a p-critical graph. If every vertex.of G is either a 6-vertex, a m-vertex or a m’-vertex, then G is an odd hole or an odd anti-hole.


107

We will prove this theorem by proving several lemmas. Throughout this section we assume the graph G satisfies the assumptions in Theorem 4. Furthermore, we only have to consider those G which contain at least one 6-vertex; otherwise Proposition 4 would apply. Also we can assume w (G) 2 4 by Proposition 1. Let u be a 6-vertex in G. Consider the unique minimum coloring of G \ u (this coloring partitions G \ u into w a-stable sets). Since N ( u ) is bipartite, at most two vertices in N ( u )can have the same color. Let M be the set of edges in N ( u ) between two vertices with the same color. Then M is an edge matching in N ( u ) . Furthermore we have the following lemma. ~

~

Lemma 2. M i s not a maximum matching. __

Proof. Suppose M is a maximum matching. Since vertices in N ( u ) which are not adjacent to a matching edge are colored singly in N ( u ) , we have, by Konig's Theorem,

-

1 maximum stable set in N ( u ) 1 = I N ( u )1 - 1 M 1 = y ( G \ u ) =w

( G\ u)= w(G).

-

Let S be a maximum stable set in N ( u ) of size w ( G ) . Thus S U {u} is a clique of size w ( G ) f 1 in G, a contradiction. 0 Given that M is not a maximum matching, there must exist an augmenting path in N ( u )with respect to M. Let P be a shortest such path. Let the endpoints of P be xo and yo with colors j and k respectively. Then colors j and k are used exactly once in N ( u ) . Let the length __ of P be 2a f 1 (a S O ) . Denote P by x o y 1 x I y 2 x 2 x r y l + l x r + l . y,x,yo in N ( u ) with colors f ( x o ) = j, f ( y o ) = k and f ( y r )= f ( x , ) = i,, 1 G 1 s a. Since P is a shortest augmenting path no xi can be adjacent to y m VO G 1 S a, m > I 1. We will show that an odd hole in G can be constructed using the path P. ~

+

Lemma 3. a 2 2 , i.e., the length of P is greater than or equal to 5. If a =0, i.e., P consists of exactly one non-matching ( x o , y o ) edge in N ( u ) , then there can be no &&)-path connecting xo and y o in G \ u (the existence of such a path would imply an odd hole containing u in G), contradicting Theorem 3. Next we consider the case where a = l . The path P is now x o y l x l y o with f ( x o ) = j , f ( y o ) = k and f(xl) = f ( y l ) = il. Consider the induced subgraph H of G on vertices colored j , k or i, and the vertex u. Clearly w ( H ) = 3. Since we assume

Proof. -

108

W.-L.Hsu

w ( G ) z = 4 , H is a proper subgraph of G. Therefore, H must be perfect and, hence, 3-colorable. This implies that there exists a 3-coloring f’ of H in which f ’ ( x l ) = f ‘ ( y o ) and f ’ ( X o ) = f ’ ( y , ) , contradicting the fact that the coloring in G \ u must be unique (by Theorem 3). 0 ___

By Lemma

a 2 2 . Hence we x o y l x I y 2 x 2 y 3.~. y.xayo ( a 3 2). 3,

Lemma 4. Each vertex in the path P

is

a

can

denote

P

(in

N ( u ) ) by

b-vertex.

Proof. We will prove the lemma by proving several claims. Claim I. xi. 0 S i s a - 1 and (let

= yo) y,, 2 G j

s a + 1 are 6-vertices

Proof. It is sufficient to prove that they cannot be f i or f i ’ vertices. (i) They are not m-vertices. A complementary graph of the one shown in Fig. 2 is shown in Fig. 4.

Fig. 4. Complement of Fig. 2.

To show that a vertex uo in G is not a fi-vertex we only have to show that in there exist vertices u l , u2, v 3 such that, together with uo, they form an induced subgraph. as shown in Fig. 4. Now, for each x,, 0 S i S a - 1 there exist x , + ~ Y, , + ~ and u such that the only edge that exists among x,, x , + ~ ,yli2 and u in G is y,+>). Hence these x,’s are not rfi-vertices. Similarly for each y,, 2 S j s a + 1 there exists Y,-~, x , - ~and u such that the only edge that exists among y,, Y , - ~ ,x , - ~ and u in G is (y,-,,x,-J. (ii) They are not m‘-vertices. Fig. 5 shows the complementary graph of the one shown in Fin. 3.

Fig. 5. Complement of Fig. 3.


109

Now, for each x,, 0 < i < a - 1 there exist x, Y , + Y~ ,,, ~ and u such that the only is (x,, Y!+~). Hence these x,’s are edge that exists among x,, Y,,~, and u in not fi’-vertices. Similar argument shows that y,, 2 < j < a + 1 are not m’vertices. 0

Proof. It is equivalent to show that they are not in G. Suppose (yl,xz)kZ E. Then we have an induced subgraph in G, as shown in Fig. 6. Now in G, xl, x2, yl, y2

it

12

Fig. 6.

must be connected by some ( i l , &)-path. Let xiQxyZ be a shortest ( i l , i2)-path connecting xI to yz. Q must contain either x2 or yl, for otherwise uxIQy2uwould be an odd hole in G. However, Q cannot contain both x2 and yl, because then ux2(along Q)ylu would be an odd hole in G. Hence assume Q contains only x2 (the case of y l is similar). Now y,xlx2(along Q)y2y3is an odd cycle in G. Since y3 is a 6-vertex, it cannot be adjacent to vertices coloied il other than xI, y l (otherwise this would give a claw). Furthermore if y3 is adjacent to any iz vertex y‘ on Q between x2 and y2 then {y3,( x l , y2, y’)} would be a claw, contradicting that y3 is a 6-vertex. Hence y3xlxz(alongQ)y2y3is an odd hole. This contradiction shows that (yl, x2)E E. By symmetry, we have (xa, ya-l)E E. 0

Claim 3. yl and x, are i-vertices. Proof. In G there exist induced subgraphs containing y,, as shown in Fig. 7. Hence y l is not a m- or fi’-vertex. Similarly we can argue that x, is not a m - or m’-vertex. 0

@-@ Fig. I .

110

w.-L.Hsu

We have shown that every vertex on the path P is a b-vertex. Now let Q 1be a shortest (i,,j)-path connecting x1 to yl in G. QI cannot contain xo, for otherwise uxo(along QI)yIuwould be an odd hole. Let yl’ be the neighbor of yl on Q1.yl’ is colored j.

Lemma 5. (yl’,yz) E E and (yl’, xl) E E. Proof. Suppose (y1’,y2) 6Z E. Then consider the induced subgraph H on vertices colored il, i2 o r j and the vertex u. Being a proper induced subgraph of G, H is perfect. Furthermore i t is easy to check w ( H ) = 3. Hence we can color H using three colors, contradicting that the coloring in G \ u is unique (by Theorem 3). Now suppose (y i, xl) 6Z E. Then in G we have a path uy;xoyIx1y2 whose edge connections are shown in Fig. 8, where u and y2 are b-vertices. Compare Fig. 8 with Fig. 6. We can apply the same argument in the proof for Claim 2 of Lemma 4 to show that Fig. 8 is impossible. Hence, ( y l , x l ) must be in E. 0

Fig. 8.

Similarly we can argue that there exists a vertex x: colored i, such that its connection in G is shown in Fig. 9. Now we are ready for the final conclusion.

Fig. 9.

Proof of Theorem 4. Following the previous discussion in this section, let us consider the odd cycle C* in G,

uy lxoylxly2(along P ) ~ , - ~ y , x . y :u. ~x

To avoid an odd hole in G, C* must contain a triangle by Lemma 1. Since N(u) is bipartite this triangle must contain either y I o r x:. Suppose there is a triangle T containing y I (the case of x: is similar); T cannot contain xl, yl or y2. Since x1 is


111

only adjacent to y l and y2 in C* the existence of such a T would imply a claw (T)} rooted at x1 in G. Hence C*contains an odd hole. By the p-criticality of G, G must be an odd anti-hole.

{x,,

It is easy to check that if we assume G is claw-free, then every vertex of G must be a 6-vertex. Hence we can obtain a proof that the SPGC is true for claw-free graphs by eliminating Lemma 4 and following the rest of the discussion in this section. This proof uses the same approach as theone in [ 7 ] . However, it is much simplified. We summarize the result in Section 2 and Section 3 as follows. Corollary 1. The SPGC is true for graphs in which each vertex is one of the following types : (i) a b-vertex, (ii) a 6-vertex, (iii) a rn-vertex, (iv) a 6 -vertex (v) a m’-vertex.

4. The SPGC on some other special graphs

We have illustrated, in previous sections, how to apply known techniques to more general graphs. In this section we show that a similar approach can produce alternative proofs that the SPGC is true for other special graphs. Those proofs turn out to be much simpler than the original ones. Corollary 2. If a p-critical graph G is planar, then G is an odd hole or an odd anti-hole. Proof. By Proposition 1, we can assume w ( G ) = 4. It is well known that a planar graph has a vertex of degree 5 or less. Let u be such a vertex in G. By Theorem 2, u is contained in 4 maximum cliques. This would imply u has degree 5 and there exist four 3-cliques in N ( u ) , which can easily be shown to be impossible. 0 A circular-arc graph is defined to be the intersection graph obtained from a collection of arcs on a circle. Characterizations of circular-arc graphs are given by Tucker [16]. The following corollary is implied by the proof of Tucker [14]. However, we will indicate the implication more explicitly here.

Corollary 3. If a p-critical graph G is circular-arc, then G is an odd hole or an odd anti-hole.

112

W.-L.HSU

Proof. We first show that there do not exist two vertices x, y in G such that (x, y ) E E and (x, z ) E E .$ (y, z ) E E. Assume such a (x, y ) pair exists. Since G is p-critical, the size of a maximum clique containing x in G \ y should be w ( G ) . But then any such maximum clique together with y would give us a clique of size w ( G ) + 1 in G, a contradiction. This implies that in a corresponding circular-arc representation of G no arc is properly contained in another arc. Thus G is, in fact, claw-free and the result follows from [12].

A toroidal graph is a graph which can be drawn on a torus so that no two edges intersect. It has been shown that the SPGC holds for toroidal graphs (see [6]). We will establish the same result using Proposition 1 and the following results of [S] and [6]. Lemma 6 [5]. Let G be a p-critical graph. If for each vertex u of G, the partition of G \ u into (Y -stable sets has at least two members containing one neighbor of u, then G is an odd hole or an odd anti-hole. Lemma 7 [6]. In a critical toroidal graph G, either w ( G )< 4, or G is regular of degree six and triangulates the torus. Corollary 4. If a p-critical graph G is toroidal, then G is an odd hole or an odd anti-hole. Proof. By Proposition 1 we can assume w(G)24. Let u be any vertex of G. Since y(G \ u ) = w(G)a4, we can color G \ u using w(G) colors, Since G is p-critical, N ( u ) must use w(G)24 colors. By Lemma 7, I N ( u ) l = 6 . Hence there must be a t least two vertices which are colored singly in N ( u ) .This is true for every vertex of G. Hence the result follows from Lemma 6. 0

References [ I ] C . Berge, Graphes et Hypergraphes (Dunod, Paris, 1970). [Z] R.G. Bland. H.-C. Huang and L.E. Trotter, Jr., Graphical properties related to minimal imperfection, Discrete Math. 27 (1979) t 1-22 (this volume, pp. 181-192). 131 V. Chvatal. R.L. Graham, A.F. Perold and S.H. Whitesides, Combinatorial designs related to the strong perfect graph conjecture, Discrete Math. 26 (1979) 83-92 (this volume, pp. 197-206). [IV. ]Chvatal, On the strong perfect graph conjecture, J. Comb. Theory, Ser. B 20 (1976) 139-141. 1.51 R. Giles and L.E. Trotter, Jr., On stable set polyhedra for K,,,-free graphs, J. Comb. Theory 31 (1981) 313326. [ h ] C.M. Grinstead. Toroidal graphs and the strong perfect graph conjecture, Ph.D. thesis, UCLA. [7] W.-L. Hsu, How to color claw-free perfect graphs, Ann. Discrete Math. 11 (1981) 18Y-197.


113

[8] H.-C. Huang, Investigations on combinatorial optimization, Ph.D. thesis, Cornell University. [9] L. LovAsz, A characterization of perfect graphs, J. Comb. Theory, Ser. B 13 (1972) 95-98. [lo] M. Padberg, Perfect zero-one matrices, Math. Program. 6 (1974) 18C-196. [l 11 K.R. Parthasarathy and G. Ravindra, The validity of the strong perfect graph conjecture for (I&\ e)-free graphs, J. Comb. Theory, Ser. B 26 (1979) 98-10, [12] K.R. Parthasarathy and G. Ravindra, The strong perfect graph conjecture is true for K,,3-free graphs, J. Comb. Theory, Ser. B 21 (1976) 212-223. [ 131 A.C. Tucker, Critical perfect graphs and perfect 3-chromatic graphs, J. Comb. Theory, Ser. B 23 (1977) 143-149. [14] A.C. Tucker, Coloring a family of circular-arc graphs, SIAM J. Appl. Math. 29 (1975)493-502. [15] A.C. Tucker, The strong perfect graph conjecture for planar graphs, Canad. J. Math. 25 (1973) 103-114.


Annals of Discrete Mathematics 21 (1984) 115-119 @ Elsevicr Science Publishers B.V.

THE GRAPHS WHOSE ODD CYCLES HAVE AT LEAST TWO CHORDS H. MEYNIEL E R 175 Combinatoire, Universifi Pierre et Marie Curie, UER 48, 7.5230 Paris, France

We prove the following theorem: If every odd cycle of length 2 5 has at least two chords, then the gruph is perfect.* This generalizes a result of Gallai and Suriinyi and also a result of Olaru and Sachs.

A k-coloring o f the vertices of G is a mapping f of the vertex set X of G into a set C of k colors such that no two adjacent vertices have the same color. A switching (relative to C ) is an operation on f defined as follows: take a connected component H of the subgraph G,, induced by the vertices of colors a, p E C. If H has at least one vertex of each color a, p, we interchange the colors (Y and p for all the vertices in H, the colors of the other vertices remaining unchanged. If H has only one vertex, of color a, say, we color this vertex with p and we color with p all the connected components of Gm,consisting of a single vertex with color a. Let 3 be the class of all graphs such that every odd cycle of length 2 5 has at least two chords, and let G E '9. For k > y ( G ) , let f be a k-coloring of G. We shall first show that it is possible to obtain from f a coloring of G with y ( G ) colors by a sequence of switchings.

Lemma 1. Let G E 9, let x be a vertex of G, and let G r ( x ) be the set of its neighbors. Consider a k-coloring of X - {x}. If y l and y 2 are two vertices of rG( x ) with distinct colors a and p, respectively, belonging to a component H of Gap,then y , and y2 are linked by an (a,P)-chain contained in rG(x). Proof. Let p [ y I y2] , = (xl = y l , xz,.. . ,x Z p = y2) be a chain of Gap of minimum length between y l and y2. The cycle p = p [ y l ,y 2 ] + [x,y l ] + [x, y z ] is odd and has only chords issuing from the vertex x. Assume that there exists a vertex a of p [ y l ,yz] which is not in rG (x); we shall show that this leads to a contradiction. * Edifors' Note. The same result was conjectured by E. Olaru (Elektron. Informationsverarb. Kybern. (EIK) 8 (1972) 147-179). Also, we have just learned from E. Olaru that S.E. Markosian and I.A. Karapotian have found independently another proof of Theorem 2.

115

H. Meyniel

116

Let x, be the last vertex in rG ( x ) which is before a on p [ y I y, 2 ] .and let x, be the first vertex in rG ( x ) which is after a on p [ y I y, 2 ] .Clearly, p [ x , ,x,] is of even length because otherwise the cycle ( x , x,, x # + ~. ., . ,x,, x ) would be an odd cycle of length 2 5 without chords. Let x k be the vertex of r G ( x ) after x, on p [ y l r y 2 ]We . have k > j + 1 , otherwise (x, x,, x , + ~. ,. . ,x,, x , + ~x, ) would be an odd cycle of length 2 5 with only one chord. As above, p [ x , , x k ]is of even length. Thus, all the portions of p [ y , ,y2] between two vertices of I‘G ( x ) are even, and therefore p [ y , .y z ] is even, a contradiction. 0

Lemma 2. Let G E %, let x be a vertex of G, and let f : X - { x } + C be a k-coloring of the subgraph G - x. Let f be the restriction o f f to the subgraph G induced by TG( x ). For every k-coloring f l of G which is obtained from f by a sequence of switchings relatively to C there exists a k -coloring g of G - x, obtained from f by a sequence of switchings relatively to C, such that g = fl. Proof. Each component H of G=@intersects G either in a single component or in several components. In the latter case, Lemma 1 guarantees that each of the several components is a single vertex. Hence, each switching on G extends into a switching or a sequence of switchings of G. 0 We remark that if GfZ 3, then the property of Lemma 2 is not always true. Consider the graph G with vertices a, b, c, d, x and edges ax, bx, dx, ab, bc, cd. The bicoloring

of G - x has a restriction

A switching on

7 gives

which is not the restriction to G of a bicoloring f l of G switchings.

-

x obtained from f by

Theorem 1. Lei G E 3, and lei f : X --+ C be a k -coloring of G. Then there exists a coloring of G with y ( G ) colors which is obtained from f by a sequence of switchings relative to C.

The graphs whose odd cycles have at least two chords

117

Proof. If the order of G is 1, then the theorem is obviously true. Let n > 1. Assume that the result is true for all graphs with order < n ; we shall show that it is true for every graph G of order n. Let C = { a I a , 2 , .. . , a k } , with k > y ( G ) = 9, and consider a k-coloring f : X + C. There exists in G a vertex x with color a # a l ,a2,. . . , aq (otherwise there is nothing to prove). Let H be the subgraph of G induced by x and the vertices of color a I ,a2,.. . ,aq.Since y ( G ) = q, the subgraph I? of H induced by rH(x) satisfies y ( H )s q - 1. By the induction hypothesis, I?, which is of order less than n and belongs to 93, has a coloring f, with y ( I ? ) colors obtained from the restriction f of f to H by a , 2 , . ., a,}. sequence of switchings relative to { a I a By Lemma 2, f l can be obtained from f by a sequence of switchings relative to { a 1a2,. , . . ,aq}.After this sequence of switchings, we can recolor x with one of the colors a l ,a 2 , . . ,aq; in other words, we can enlarge the set of vertices colored with a l ,a z ,. .. , aq.This process can be repeated until all the vertices of G are colored with a I , a z , .. . , aq.Thus we obtain a coloring of G with y ( G ) colors. 0 Theorem 2. Let G be a graph such that every odd cycle of length more than 3 has at least two chords. Then G is perfect. Proof. Consider a graph G E 3 which is not perfect and of minimal order. So, if q = y ( G ) , we have w ( G ) < y ( G ) = 9 ; furthermore, for every set A C X , A # X ,

7

Let x be a vertex of G ; so the subgraph G - x has a ( q - 1)-coloring f . Let be the restriction of f to the subgraph G induced by r G ( x ) . We have y ( G ) 3 9 - 1: otherwise, by Theorem 1, G has a (q -2)-coloring f l , which is obtained from f by switchings; so, by Lemma 2, there exists a (9 - 1)-coloring g of G - x with g = Hence y ( G ) C 9 - 1, a contradiction. Let A = { x } U rG (x), then y ( G A )3 q ; so, by (l),A = X , and x is adjacent to every other vertex. Since x was chosen arbitrarily, this shows that G is a clique. Hence y ( G ) = w ( G ) , and the contradiction follows. 0

TI.

Corollary 1 (Gallai [3]). Let G be a graph such that every odd cycle of length > 3 has at least two non-crossing chords; then G is perfect. Corollary 2 (Olaru, Sachs [6]). Let G be a graph such that every odd cycle of length > 3 has at least two crossing chords ; then G is perfect.

I in

H. Meyniel

Corollary 3 (Hajnal and Suranyi [4]). Let G be a triangulated graph (that is, every cycle of length > 3 has a chord). Then G is perfect. Proof. I n this case every odd cycle of length theorem follows. 0

3 5 has

at least two chords and the

Corollary 4 (Seinsche [7]). If a graph G has no induced P4(elementary chain with -1 verlices and no chords), then G is perfect. Proof. Every odd cycle of length at least 5 and with at most one chord contains an induced P,. 0 Corollary 5. I f a graph G has no induced subgraph isomorphic to Go= k > 1. (nb, ac, bc, c d } , then G is perfect if and only if it has no induced Proof. A graph G with no CZk+, and no Go has no odd cycle with only one chord: the unique chord would be triangular, which would yield an induced Go. 0 Remark. The method used in Theorem 2 cannot yield a proof of the perfect graph conjecture. For example, consider the graph G with vertex set {x, : 1 s i G 3 , 1 S j S 4}, where {x,,x,} is an edge iff i f p and j f q. The graph G is perfect (since its complement has a clique-hypergraph which is balanced, and, by a theorem of Berge [2, Chapter 161, this implies that this complement is perfect). Clearly. G E 3, and from the 4-coloring with classes S, = {x,] j = 1,2,3) we cannot get a 3-coloring by a sequence of switchings.

I

Problem. We denote by C , ( G ) the set of colorings of G with q colors, 1,2,. . . ,q. We say that two colorings f , g E C,(G) are q-Kempe-equiualent if there exists a sequence of colorings all in C, (G), f n = f , f l , . . . ,f k = g, such that for i = 1,2,. . . , k, the coloring f, is obtained from f,-, by a switching. The Kempe-equivalence being an equivalence relation, the classes of this relation will be called the q-classes of G. If G is the graph with vertex set {1,2,3,4} x {1,2,3,4,5} x{1,2,3,4.5}, where the vertices ( i , j, k ) and ( i ' , j', k ' ) are adjacent iff i # i f ,j # j ' and kf k', we make the following observations: (1) The projection p l ( i , j , k ) = i is a 4-coloring of G ; the projections pz(i, j, k ) = j and ps(i,j, k ) = k are 5-colorings. (2) For q 3 4, every q-coloring of G is q-Kernpe-equivalent to pl, or to p2,or to

p3.

(3) The colorings pl, p 2 and p3 belong to three different 6-classes.

The graphs whose odd cycles have at least two chords

119

(4) The coloring p 2 is q-Kempe-equivalent to pl for q 7. Thus, the number of q-classes is 1 for q = 4 ; it is 3 for q = 5 or 6; it is 1 for q 3 7. This suggests the following open problem: If in G every odd cycle has at least two chords, is the q-class unique for every q y ( G ) ?

References [I] C. Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. 2. Matin-Luther-Univ. Halle-Wittenberg (1961) 114. [2] C. Berge, Graphes et hypergraphes (Dunod, Paris, 1970); English translation: Graphs and Hypergraphs (North-Holland, Amsterdam, 1973). [3] T. Gallai, Graphen mit triangulierbaren ungeraden Vielecken, Magyar Tud. Akad. Mat. Kutato Int. Kozl. 7 (1962) A 3-36. [4] A. Hajnal and J. Surinyi, Uber die Auflosung von Graphen in Vollstandige Teilgraphen, Ann. Univ. Sci. Budapestinensis 1 (1958) 113. [5] M. Las Vergnas and H. Meyniel, Kempe classes and the Hadwiger conjecture, J . Comb. Theory. Ser. B 31 (1) (1981) 95-104. [6] H. Sachs, On the Berge conjecture concerning perfect graphs, Combinatorial Structures and their Applications (Gordon and Breach, New York, 1970). [7] D. Seinsche, On a property of the class of n-colorable graphs, J. Comb. Theory, Ser. B 16 (1974) 191-193. [8] L. Surinyi, The covering of graphs by cliques, Studia Sci. Math. Hungar. 3 (1968) 345-349.



CONTRIBUTIONS TO A CHARACTERIZATION OF THE STRUCTURE OF PERFECT GRAPHS Elefterie OLARU and Horst SACHS Technische Hochschule Ilmenau, DDR -6300 Ilmenau, German Dem. Rep.

Characterizing the structure of perfect graphs is one of the important, most actual, and most challenging unsolved problems of graph theory. The strong version of Claude Berge’s Perfect Graph Conjecture has given rise to numerous investigations of this problem. In this paper, results found by E. Olaru since 1969 are summarized.

1. Introduction

The graphs G = ( X , U ) considered in this paper are finite, undirected, and have neither loops nor multiple edges; X and U denote the sets of vertices and of edges, respectively. We shall use the following general terminology and notation. S ’ C S means: S’ is a subset of S ; S ’ C S means: S‘ is a proper subset of S. Let X ’ C X , U’ 5 U ; G = ( X , U ) , G’ = ( X ‘ ,U’). The graph G ’ is called a partial subgraph of G ; if any two vertices of X ’ are adjacent in G ’ if and only if they are adjacent in G, then G‘ is called a subgraph of G or, more precisely, the subgraph of G spanned by X ‘ , and denoted by [ X ’ ] ,;if X ’ = X then G ‘ is called a partial graph of G. The number of vertices of G , i.e., the cardinality I X I of X , is also denoted by n(G). A complete subgraph of G is called a clique of (or in) G . The maximum number of vertices contained in a clique of G, i.e., the ‘density’ of G, is denoted by w ( G ) . A subset S C X with [SIC= (S,@ is called a set of independent vertices or, briefly, an independent set of G . The maximum number of independent vertices of G is called the stability number of G and denoted by a ( G ) .Here, a stability system of G is an independent set S of G with 1 S I = a ( G ) . The complement of G = ( X , U )is the graph G = ( X , 0) with the property that any two vertices of X are adjacent in G if and only if they are non-adjacent in G . Clearly,

a ( C )= w ( G ) ,

w ( G )= a(G). 121

(1)

E. Olanc, H. S a c k

122

For X' C X and U ' C U, we define G-X'=[X-X']c,

G-U'=(X,U-U');

f o r x E X , u E U , w e w r i t e G - x a n d G - u insteadof G - { x } o r G - { u } , respectively. Let G ' be a subgraph of G ; an edge u of G is called a-critical with respect to G ' if

a(G'-u)=a(G')+l.

(2)

The set of all edges of G which are a-critical with respect to G itself is denoted by U', and the partial graph G' = (X, U c )of G is called the &-critical skeleron of G. If G = G' then G is called a-critical.

2. a -partitionable graphs

Let G = (X, V), let Z = {XI,X,,. . .,X m }( m a 1) denote an arbitrary partition of X (i.e., X = XI u X1u . . u Xm,X, # 0, Xi n X, = 0 ( i # j ;i, j = 1,2, . . . ,m)), and put G, = [X,Ic ( i = 1,2,. . . ,m ) . Then, evidently,

-

m

a ( G ) sT a ( G t ) , I =

(3)

m

w(Gi).

w(G)s I=

Definition 1. The graph G = (X, V ) is called a -partitionable if a ( G )= 1 or if there is a partition Z = {XI,X2,.. . ,X,} of X with m > 1 such that m

Z is then called an a-partition of G, and the graphs Gi are called the corresponding a -components of G. G is called w-partitionable if G is a-partitionable. G is called parfirionable if it is both a-partitionable and w-partitionable.

Of particular interest in graph theory are those a -and w-partitions for which all G,have a certain prescribed property. An an example, consider the following unsolved problem: Characterize all graphs G having one of the following two properties: (C) G has an a -partition with all a -components being cliques; (F) G has an w-partition with no w-component having an edge.

Structure of perfect graphs

123

Remark. If the graph G = ( X , U )has an a -partition Z such that the corresponding a-components are cliques then, obviously, the number of these components equals a (G), i.e., G can be covered by a ( G )cliques. We shall then say: G has a perfect clique -covering or G is perfectly a -partitionable.

z

If G has an o-partition 2 satisfying (F) then generates a colouring of X with exactly w ( G ) colours. We shall then say: G has a perfect colouring or G is perfectly w -partitionable. Definition 2. A graph G is called a-perfect (or w-perfect) if each subgraph of G (including G ) has a perfect clique covering (or a perfect colouring, respectively). G is called perfect if it is both a-perfect and o-perfect.

Before dealing with perfect graphs, we shall prove some general statements concerning a -partitionable graphs. Remark. If G is disconnected G is a-partitionable, for, if G has C1, C2, . . . ,C,,, ( m > 1) as its components of connectivity, then, clearly, rn

)). a ( G )= i = l a (C;

Next we prove Theorem 1. Let G = ( X , U ) be a connected graph and assume that there is a set Q C X such that (i) G - Q is disconnected, (ii) [a], is a clique, then G is a-partitionable. Proof. If

u(G - Q ) = a(G)- 1

(5)

then G is a-partitionable since (5) implies a ( G )= a ( G - Q) + a ([ a],). Therefore we assume a(G - Q)= CY(G).

(6)

Because of (i), G - Q is the union of two (non-empty) separated parts GI= (XI,Ul)and Gz = (X2,U2).The set of the stability systems of G - Q is obtained by combining in every possible way a stability system of GI with a stability system of G,; thus, because of (6), we have

(GI)+ (G2)= ( G ) .

(7)

E. Olaru, H.Sachs

124

We shall now show that Q can be partitioned into two parts a way that

a ( [ X ,U Q,IC)= a ( [ X , I c ) =a ( G ) ( i where Q1or

Q2

Ql

= 1,2)

and Q2in such (8)

may be empty. From (8) and (7) we deduce

..([Xi U Q,]C)+(Y([X~U Q,]~)=~~(G~)+(Y(G~)=(Y(G), i.e., G is a-partitionable. Let x be an arbitrary vertex of Q. We shall show that either in every stability system of GI or in every stability system of G2there is a vertex which is adjacent to x. For suppose that S1 is a stability system of GI,say, with none of its vertices adjacent to x : then, if S2 is an arbitrary stability system of G1, x is adjacent to at least one vertex of S2, for otherwise there would exist a stability system S = S , U Sz of G with none of its vertices adjacent to x - contradicting the definition of a stability system. Now let QIcQ be the set of all those vertices of Q which are adjacent to at least one vertex of every stability system of GI:then every vertex of Qz= Q - QI is adjacent to at least one vertex of every stability system of Gz.Clearly, Ql and Q2 satisfy (8), which proves the theorem. 0

A

For an arbitrary graph G = (X, V )and sets A C X, B C X, where A # 0,B # 0, n B = 0, we define

[A,Blc = { u ( u E U ; u = ( a , b ) , a € A , b EB}. Theorem 2. Let G be an a -partitionable graph and let {XI,Xz, . . . ,X,,, } be an a-partition of G. Then none of the sets [X,,XjlC (i# j; i, j = 1 , 2 , . .. ,m ) contains an a-critical edge of G.

Proof. Assume that u = (a, b) is an a-critical edge of G. Then, by definition, G - u has a stability system S with IS I = a ( G ) +1. From

/ s / = a ( G ) + i= l S n x l =

i=l

m

= C Isnx,~ i=l

1

we deduce that there is an io such that S f l X,I

a([X,]c-.)=ff(G,)+

> a(G,). This implies that

1,

which is impossible unless a, b EX,.

0

Corollary 1. lf G is (Y -partitionable and

(Y

( G )> 1 then the a-critical skeleton G'


125

of G is disconnected and each connected component of G' is contained in an a -component of G. Corollary 2. Each a-critical edge of an a-partitionable graph G belongs to an a-component Gb of G and is a-critical with respect to G,. Corollary 3. A n a -critical graph G with a (G) > 1 is a -partitionable if and only if G is not connected. Note that there are a -non-partitionable graphs whose a -critical skeletons are disconnected. As an example consider G = ( X , U ) where X = {xl,x2, x3, x4,XS, x ) and

U = { X I , XZ),

(x2, x3), (x3, x4), (x4,

u {(x, x i ) 1 i = 1,2,3,4,5}

xs), (xs, XI))

(the "5-wheel"):

here G' = (X,V c )with U' = {(XI,XZ), (XZ, x3), (XS, x4), (x4, XS), (XS, XI)), thus G' is disconnected. Clearly, any graph G = ( X , U ) contains a partial graph G' = ( X , U ' )(U'C U ) with the following properties: (a) a ( G ? = a (GI; (b) every edge of G' is a-critical with respect to G'. G ' is called an a-critical partial graph of G. Theorem 3. A graph G with a ( G )> 1 is a-partitionable if and only if it contains at least one a-critical partial graph which is disconnected (i.e., by virtue of Corollary 3 of Theorem 2, which is a -partitionable). Proof. First suppose that G is a-partitionable; let Z = {Xl,Xz,.. .,X m }(rn > 1) be an a-partition of G. If there is, for some i €{2,3,. . ., m } , an edge u1€[XI,X,], such that a ( G - u ; ) = a ( G ) , then u I is omitted and G - u1 is denoted by GI. Clearly, Z is an a-partition of GI. If there is an edge u2E (XI,Xi),, such that a ( G 1- uz)= a (GI)= a ( G ) , then uz is omitted and GI - u2 is denoted by Gz, and so on. After a finite number of steps we arrive at a partial graph G, of G having the following properties: (i) a ( G p) = a ( G ) ; (ii) Z is an a-partition of G,; (iii) every edge contained in lXl,X,]Gp is a-critical with respect to G,. Corollary 2 of Theorem 2 now yields that [XI,X , ] , is empty, which implies that not every a-critical partial graph of Gp can be connected. If G contains a disconnected a-critical partial graph G' then, according to Corollary 3 of

E. Olaru, H.Sachs

126

Theorem 2, G' is a-partitionable. An arbitrary a-partition Z = {XI,X2,. . . ,X k } of G' is an a-partition of G, too, since

Theorem 3 is now proved.

0

Definition 3. A graph G is called strongly stable if for each clique C = [a], of G, a ( C - Q ) = a ( G ) . Remark 3. If G is a-non-partitionable then G is either strongly stable or complete. The converse of this statement, however, is not true: the graph consisting of two separated 5-circuits is a -partitionable though it is strongly stable. But the following theorem is valid: Theorem 4. If a graph C is strongly stable and a-partitionable, then all a components of any a-partition of G are strongly stable. Proof. Let G = (X, V )be a strongly stable graph and let Z = {XI,X,, . . . ,X,,} be an arbitrary partitlon of X such that

Assume that one of the a-components, say GI,is not strongly stable. Then there is at least one clique C = [QlC, of GI satisfying a(G1-

a)= a(C1)- 1.

(10)

Since {XI - Q, X 2 , .. . ,X.} is a partition of X - Q, (9) and (10) imply

( G - Q) C (GI- Q) + (Gz) +

*

+

( G , ) = ( G )- 1.

But this inequality contradicts the hypothesis that G is strongly stable.

0

As a simple consequence of Theorem 4 we have

Corollary 1. Let G = (X, U )be a strongly stable graph. Then there is a partition {XI,X z , . . . ,X,} (p 2 1) of X such that (a) all G, = [XI,(i = 1,2,. .. ,p ) are strongly stable and a -non-partitionable, (b) a ( G , )= cu(G).

xr=l

Remark. We shall see that, in the investigation of perfect graphs, the strongly stable graphs are of a particular significance.


127

3. Perfect graphs Let x ( G ) denote the chromatic number of the graph G. Obviously,

X ( G ) Zw ( G ) and, analogously, x(G)Zw(G)=a(G). For many purposes the following reformulation of the definition of perfect graphs (see Definition 2) is more appropriate.

Definition 2’. The graph G is called (a) a-perfect if = a ( G ‘ )for every subgraph G’ of G ( G ’ = G included); (b) w-perfect if x ( G ’ )= w ( G ’ )for every subgraph G‘ of G (G’ = G included); (c) perfect if G is both a-perfect and o-perfect.

~(c’)

Note that, in fact, properties (a), (b), and (c) are equivalent: this is the content of the weak version of Berge’s Perfect Graph Conjecture and follows easily from an important result of L. L O V ~ S[lo], Z [ll]:

Theorem 5 (Lovisz [lo]). A graph G is perfect if and only iffor every subgraph G ‘ = [X’], (X’CX)

n ( G ’ ) = I X ’ l s a ( G ’ )w . (G’). Remark. x ( G )equals the minimum number of cliques covering the vertex set of G : therefore, x ( G ) is also called the clique-covering number of G denoted by O(G). We need some notions and results. 3.1. O d d holes and

(Y

-critical Hamiltonian circuits

The graph G = (X, U )is called a k-circuit (of length k 3 3) and denoted by L if X can be so arranged as a sequence xl, x 2 , . .. ,X k that

u = {(Xi,Xi+])

I i = 1,2,. . . ,k ;

Xk+1

k

= XI}.

We shall briefly write L k

=( x l ,

x2,. ,

.

9

X k , XI)

and say that xi and xi+]( i = 1,2,. . . , k ; X k c l = xl) are neighbouring on L k . If the circuit L is contairred in a graph G (as a partial subgraph) we shall say that L is a

E. Olaru. H. Sachs

128

circuit of (or i n ) G. An edge of G connecting two vertices of a circuit L of G which are non-neighbouring on L is called a chord of L. Let s = (x, y ) and s‘ = (x’, y’) be two chords of a circuit L in G where the vertices x, y, x’, y’ are pairwise distinct. If, on L, the pair x’, y ‘ separates (and is separated by) the pair x, y we shall call s, s f a pair of crossing chords of L.

Definition 4. A circuit of a graph G of length greater than three which has no chord in G is called a hole of G. Towards a characterization of the structure of &-perfect graphs the following conjecture has already become famous.

Conjecture 1 (C. Berge and P.C. Gilmore) (strong version). A graph G is a-perfect if and only if neither G nor its complement G contains an odd hole. For the sake of brevity we introduce the following notation: Let G = ( X , U ) be an arbitrary graph, let X ‘ , X ” C X with X ’ n X ” = 0;we write X’pcX” if in G every vertex of X ’ is adjacent to every vertex of X ” , and we write X’pCX“ if there is in G no edge connecting a vertex x f E X ’ with a vertex x “ E X ” ; instead of { x } p c Y we briefly write xpcY, etc. The next theorem yields a necessary and sufficient condition for the existence of an odd hole. Theorem 6. A graph H = ( X , U ) contains an odd hole i f and only if there are in H two distinct edges ( x ’ , x ) and (x,x’) such that (i) ( x I , x ) and (x,x2) are a-critical with respect to some subgraph G of H ; (ii) ( x I r x Z ) U. ~ If H satisfies the above condition then there is in H an odd hole L which contains both edges (x’, x ) and ( x . x’). Proof. 1 ( “ i f ” ) : Because of

a ( G - ( x ’ , x ) ) = a ( G ) + 1 ( i = 1,2), there are two stability systems S , and S2 of G such that x ’ E s,,XP xpCS,

-{XI}

s,, ( i = 1,2).

Now consider the subgraph G‘ of G spanned by SIU Sz U {x}. We have a ( G ’ ) =a ( G ) ,

(11)


129

a ( G ’ )< x(C’),

(12)

o(G’)=2.

(13)

The first relation is clear since, by definition, I S, I = a ( G ) . Expression (11) implies that (x’, x ) and ( x , x2) are a-critical also with respect to G ‘ . If there is an a-partition of G ’ with all its a-components being cliques then, by Corollary 1 of Theorem 2, the a-critical edges (x’, x ) and (x, x2) are contained in a clique of G ‘ , i.e., ( X I , x2) is an edge of H, contradicting (ii); thus (12) follows. The subgraph spanned by S, U Sz is bipartite, thus using (11) and (ii) we conclude that G ‘ does not contain a 3-circuit; therefore, w(G’)= 2 , i.e., (13) is valid. Since a bipartite graph is a-perfect, (12) and (13) imply that G ’ contains an odd hole, say L * . Necessarily, ( x l , x ) and (x,x2) belong to L* for G I - ( x , x ’ ) ( i = 1,2) are bipartite graphs. 2 (“only if”): If H contains an odd hole L then any two adjacent edges ( x ’ , x ) and (x,x*) of L are a-critical with respect to L and, clearly, satisfy (ii). Theorem 6 is now proved. Corollary. A n a-critical graph is a-perfect if and only if all of its connected components are cliques. Definition 5. A Hamiltonian circuit of a graph H which has the property that all of its edges are a-critical with respect to H is called an a-critical Hamiltonzan circuit of H. From Corollary 1 of Theorem 2 we deduce the following statement: A graph H which contains an a-critical Hamiltonian circuit and which is perfectly a -partitionable is a complete graph.

This implies the subsequent statement:

A graph H with a(H)2 2 which contains an a-critical Hamiltonian circuit cannot be perfectly a -partitionable. Definition 5a. A subgraph G of an arbitrary graph H is called an a-critical hole of H if (i) a ( G )3 2, (ii) G contains an a-critical Hamiltonian circuit. Remark. A graph H which contains an a-critical hole cannot be a-perfect.

E. Ohm, H. Sachs

130

Conjecture 2 (see [17]). A graph H is perfect if and only if it has no a-critical hole. Remark. An odd circuit holes.

L2kCl

(k 3 2) and its complement

z2k+l

are a-critical

3.2. Critically a -imperfect graphs

For the investigation of perfect graphs we have introduced the notion of a critically a-imperfect graph and we have shown (see [13]) that, using this notion, many propositions concerning perfect graphs - new ones and also well-known ones - can be proved in a unified and relatively simple manner. We hold that the notion of a critically a-imperfect graph plays a particular role within the theory of perfect graphs and, therefore, believe that it is useful to continue the investigation of critically a -imperfect graphs until a reasonable structural characterization of perfect graphs is arrived at. (See also the Concluding Remark of this paper.)

Definition 6. A graph G is called critically a-imperfect if (A) 8 ( G ) > a(C) (recall: 8 ( G ) = x ( G ) ) ; (B) e(Cr)= a ( G r )for every proper subgraph G' of G.

Lemma 1. Every a -imperfect graph contains a critically a-imperfect subgraph.

Proof. (Clear.) Lemma 2. A graph H is a-perfect if and only if it contains no strongly stable subgraph.

Proof. If H contains no strongly stable subgraph then every subgraph H' of H (including H)has a clique-covering of exactly a (H')cliques, i.e., H is perfect. To finish the proof just note that a strongly stable graph G cannot be partitioned into a ( G ) cliques. 0 There is a tight relation between strongly stable and critically a -imperfect graphs, to be expressed in the next theorem. First we give the following definition: Definition 7. A strongly stable graph which contains no strongly stable proper subgraph is called a minimal strongly stable graph. Now we prove the next theorem.


131

Theorem 7 . A graph G = ( X , U ) is critically a-imperfect if and only if it is a minimal strongly stable graph. Proof. If G is critically a-imperfect then it is a-non-partitionable (and, consequently, strongly stable) for otherwise there would exist an a-partition, say { X , ,X z } , with a ( G J a ( G Z = ) a ( G ) . The graphs G I and Gz being proper subgraphs of G, G Ican be covered with a ( G I )and G2can be covered with a ( G 2 ) cliques, thus G can be covered with a ( G )cliques -contradicting the hypothesis that G is critically a-imperfect. From Lemma 2 it follows that G (since it is critically a -imperfect) contains no strongly stable proper subgraph: this proves that G is a minimal strongly stable graph. Conversely, let G be a minimal strongly stable graph. Then, by Lemma 2, G is a-imperfect. Now, since G contains no strongly stable proper subgraph, again by Lemma 2, every proper subgraph of G is a-perfect. Theorem 7 is now proved. 0

+

Before L O V ~ Spublished Z his important result (Theorem 5), we established a couple of properties of critically a-imperfect graphs without using Theorem 5 (see [13]-[18]). In making use of Theorem 5 we shall now simplify the original proofs.

Theorem 8. A graph G = ( X , U ) is critically a-imperfect if and only if (i) I X I = a ( G ) * w ( G ) + l , (ii) each proper subgraph G’ = [ X ’ ] , ( X ’ C X ) of G satisfies

I X ’ 1G a (G‘) w (G’). *

Proof. Clearly, (i) and (ii) imply that G is critically a-imperfect (see Theorem 5). Conversely, if G is critically a-imperfect then Theorem 5 implies that (ii) is valid and that

1 X I S a ( G ) *w ( G ) + 1. Now, let x be an arbitrary vertex of G. Then, by virtue of Theorem 7 and property (B) of G (see Definition 6), e ( G - X) = a ( G )and, since a clique cannot have more than o ( G ) vertices,

n ( G - x ) = I X I - 1 G a ( G )* w ( G ) . Thus (i), too, is valid.

17

Corollary. A graph G is critically a-imperfect if and only if its complement G is critically a -imperfect.

E. Olary H. Sachs

132

Let C = (X,U ) be critically a -imperfect and let x be an arbitrary vertex of G. Because of property (€3) of C (see Definition 6), G - x is a -partitionable and Theorem 7 implies a ( G - x ) = a ( G ) . Let C,, C,, . . .,C, be an arbitrary system of cliques covering G - x and satisfying

c, = [X,],x, , cx - { x } = X,nxl=O

X,#O,

n

u x,,

,=I

( i # j , ; i , j = l , 2 ,..., a ) .

G X , ( i = 1,2,. . . ,a)denote the set of all those vertices of X, Further, let X : which, in C, are non-adjacent to x, i.e., such that x&X!

and

x p c ( X , - Xf).

Then we have the following lemma:

Lemma 3. For every x E X and for every perfect a-partition of G - x, the following propositions hold : (1) for i = 1,2,. . . , a the set X fis non-empty, ( 2 ) for eoery ser of subscripts J C{l, 2 , . . . ,a } = I, the equality

holds.

Proof. (1) If there is an i with X : = 0 then C : = EX,U {x}IC is a clique and the cliques C , ,. . .,C,-,, C:, C,,,,. . . , C, generate a perfect a-partition of G, contradicting property (A) of critically a-imperfect graphs (see Definition 6). ( 2 ) The graphs [Xi],being cliques,

immediately follows. Let us assume that there is a set of subscripts J C I (i-e.,I J 1 < (Y (C)) such that a([

u x:] )SIJl-l.

IEJ

C

This assumption implies

ujGJX,

U { x } l C either contains x - then, by our since a stability system of [ assumption, it cannot have more that I J I vertices - or it does not contain x -


133

then, again, the system cannot have more than ( J i vertices since the graph [ j E J X , ] G is partitioned into exactly I J I cliques having pairwise disjoint vertex sets. The graph [ U,,,X], is covered by a ( G ) - ( J Jcliques, therefore

u

a(

[ u XI iGJ

G

)Ga(G)-IJI.

ujEIX,

u,,JX,]G

The relation I J I < a ( G ) implies that [ U {x}], and [ are proper subgraphs of G , and using property (B) (see Definition 6) we conclude that

B(G)= B(

[ u X, U { x } U u X i ] ) j€J

is1

G

contradicting (A) of Definition 6. Lemma 3 is now proved. 0 As an easy consequence of Lemma 3, we obtain the following corollary:

Corollary. (1) Every vertex of a critically a -imperfect graph G is contained in at least a(G)(pairwise distinct) stability systems of G . (2) A critically a-imperfect graph G has at least a(G)*w ( G ) + 1 (pairwise distinct) stability systems. For every vertex x EX, we define the following sets:

I xz= {xz I x z E ( X - {x}) and x 2 p c x } , X ' = { x i x 1 E ( X - {x}) and x ' p c x } ,

and the corresponding graphs:

G t x ) =[XI],, G : x ) = [ X 2 ] ~ . Lemma 4. The graph Gtx1is perfectly ( a ( G ) - 1)-partitionable, and each clique of any perfect ( a ( G ) - 1)-partition of G f , ) contains at least two vertices. Proof. Lemma 3 , part ( 1 ) yields

Lt

x i =U x:,xt#0, x:nx:=0, i-I

and the graphs [ X l ] , are cliques ( i # j ; i,j = 1,2,. . . , a ) . From Lemma 3, part ( 2 ) we obtain a ( C i X ) ) ab ( G ) - 1. If there is a stability system S ' of Gll, with 1 S' I > a ( G ) - 1 then, because of x&X', the set S' U { x }is a stability system of G with S'U { X I 1 > a ( G ) ,contradicting the definition of a stability system. Consequently, a(Gi,))= a ( G ) - 1; the graph Glx, being a proper subgraph of G, property (B) (see Definition 6) implies

I

e(G{x))= a(Cf,)). In order to establish the second part of the assertion, we first prove

(**I

a ([XI - { x ' } ] ~=)(Y ( G )- 1 for every x I E XI. By virtue of (*), X I belongs to precisely one of the sets X l , say part ( 2 ) of Lemma 3 we obtain

X I

E X : i ;from

( ~ ( [-xXii],.) ' = ( Y ( G ) -1. Now,

a ( [ X '-xli]c)s Cr([X'-{x'}]c) (Y([X']G) = a(C[,,)= a ( G ) - 1,

hence (**) is valid. Let us assume that there is a perfect ( a ( C )- lkpartition o f G:x)such that one of the cliques of this partition consists of a single vertex, say X I . Then the graph [ X ' - { X ' } ] is ~ covered by a ( G ) - 2 cliques,.thus ( Y ( [ X ' - { X ' ) ] ~ ) ~ ( Y ( G ) - ~ , contradicting (**). Lemma 4 is now proved. 0

Corollary 1. Let G be a critically a-imperfect graph, let x be an arbitrary vertex of G, and denote the valency of x with respect to G.by r ( x ; G ) . Then r ( x ; G ) sn ( G ) - 2 a ( G ) + 1. Proof. By Lemma 4, I X1I k 2a ( G )- 2, thus

r ( x ;C ) = ) X z = ( n ( G ) - I X ' I - 1 S n ( G ) - 2 a ( C ) + 1. 0 Corollary 2. For any critically a-imperfect graph G,

+

n ( G )2 2 a ( G ) 20(G)-3.

(14)


135

Proof. Together with G, its complement G is also critically a-imperfect (see Theorem 8), hence, by Lemma 4,

J X'

(***I

13 2 a ( G ) - 2 = 2 w ( G )- 2;

thus n ( G ) = 1 X'

1 + I X 2I + 1 3 2a (G)+ 2 w ( G ) - 3. 0

Remark. We shall show (Theorem 11) that equality holds in (15) if and only if the critically a-imperfect graph G is an odd circuit LZq+'or the complement LZq+' of an odd circuit (q 3 2). To prove this, we need some more results. Lemma 5. Every vertex x of a critically a-imperfect graph G is contained in an odd k-circuit ( k 3 5 ) L of G such that all chords of L (if there are any) are issuing from x. Proof. Again we consider an arbitrary vertex x of G and the corresponding subgraphs G i I )= [X'],and G:=)= [X'],.By Lemma 4, a ( G t X )=) a(G)- 1. Let X 2be a subset of X 2satisfying

a ( [ X ' U PI,)= a ( G )- 1 and having, under this condition, the maximum number of vertices:

a([xlu X' u { y ) ] , ) = .(G) for every y E (x' - P). (16) [X' U PI, is a proper subgraph of G (since x e X' U X2), therefore, because of property (B) (see Definition 6),

The set X 2- X 2does not span a clique in G for, otherwise, [(X'- X') U {x}], would be a clique and using (17) we should obtain O(G)= a ( G ) , contradicting property (A) (see Definition 6) of critically a-imperfect graphs. From this and from (16) we conclude that there exist two vertices XI,x' E ( X 2- X') such that xpc{x1,xZ1, x ' p c x 2 ,

(18)

and

a ([XI U X2U {x '}Ic)

= a ( G)

(j= 1,2).

(19)

and denote the Now we omit all edges of G which connect x with vertices of remaining graph by H. We have a ( H )= a ( G )since a stability system of H that does not contain x is a stability system of G as well, and a stability system of H that contains x has,

136

E. Olaru, H. Sachs

because of (17) and because of x p H ( X z- X') and x p H ( X 'U X'), exactly a ( G ) vertices. From (18) and (19) we obtain just those conditions formulated in Theorem 6 (with respect to H), thus H contains an odd hole L passing through x. To such a hole of H there corresponds in G an odd k-circuit ( k 2 5) passing through x and having the property that all its chords (if it has any) issue from x (namely, these chords are precisely those edges of G which connect x with those vertices of X2 that lie on L). x, having been chosen arbitrarily, Lemma 5 is now proved.

Remark. The following proposition (see [13], Theorem 10) is also valid: Proposition. Every edge (x,y ) of a critically a-imperfect graph is contained in an odd k-circuit ( k 2 5 ) all of whose chords (if there are any) are issuing from x, and, simultaneously, in an odd k'-circuit ( k ' 3 5 ) all of whose chords (if there are any) are issuing from y . Corollary 1. If the critically a-imperfect graph G is not an odd circuit Lk ( k 3 5 ) then every vertex x of G is contained in a 3-circuit. 0 Proof. According to Lemma 5, every vertex x of the critically a-imperfect graph G lies on an odd circuit L of length 2 5 which, except for those (possibly existing) chords issuing from x, has no chords. The critically a-imperfect graph G is not an odd k-circuit with k 3 5, hence the vertex set of L spans a subgraph of G which contains at least one 3-circuit (since, otherwise, G would contain an odd hole as a proper subgraph). Evidently, x belongs to this 3-circuit.

Corollary 2. If every odd k-circuit ( k 3 5 ) of a graph H has a pair of crossing chords then H is a-perfect. Proof. Suppose H were not a-perfect. Then, by Lemma 1, H contains a critically a-imperfect subgraph G. Lemma 5 implies that every vertex of G lies on a circuit of length 2 5 without crossing chords. This contradicts the hypothesis of the corollary. 0

Remark. In 1968 (see [16]), we conjectured that the existence of two chords in every odd k-circuit ( k 2.5)of H implies the perfectness of H. In 1976, Olaru [17] proved the following statement:

If a graph H satisfies the following conditions:


137

(i) every odd k-circuit ( k 3 5 ) of H has at least two chords, (ii) no 5-circuit of H has exactly one pair of crossing chords, then H is a-perfect. Only recently, Markosjan and Karapetjan (see also [12]) proved the abovementioned conjecture:

Theorem 9 (Olaru, Markosjan and Karapetjan). If every odd k-circuit ( k 2 5 ) of a graph H has at least rwo chords then H is perfect. Theorem 10. A critically a-imperfect graph G is (1) an odd circuit L 2 k + l ( k 3 2) if and only if o(G)= 2; (2) the complement of an odd circuit L Z k + l ( k 3 2) if and only if a ( G )= 2.

Proof. (l)(a) First we show: An odd circuit L 2 k + ) ( k 2 2 ) is critically a imperfect. For every vertex x of L Z k + l , the graph L2k+l - x is bipartite and, therefore, a-perfect. L 2 k + l itself, however, is not a-perfect for n ( L Z k +=l )2k + 1, a ( ( L Z k + l ) = k, and @ ( L Z k + l ) = 2 imply 8(L2k+l)> a (L2k+l); thus L 2 k + l is critically a -imperfect. (b) Now we assume that G is critically a-imperfect with o(G)= 2. If G is not an odd circuit then each vertex of G lies on a 3-circuit (see Corollary 1 of Lemma 5), contradicting w ( G ) = 2; thus G must be an odd circuit. (2)(a) First we show: The complement L Z k + l of an odd circuit L Z k + l (k 2 2) is critically a -imperfect. For every vertex x of L Z k + l , the graph i 2 k + l - x is the complement of a bipartite graph; therefore, each proper subgraph of L 2 k + I is perfectly a partitionable (and hence a -perfect). & k + I itself, however, is nor a-perfect for n(LZk+1)=2k 1, ( Y ( L 2 k + 1 ) = 2 , and @ ( L 2 k + I ) = ( Y ( L Z k + l ) = k imply that 8 ( L Z k + I ) > a ( L 2 k + I ) ; thus E 2 k + l is critically a-imperfect. (b) Now we assume that G is critically a-imperfect with a ( G )= 2. Suppose of G does not contain an odd hole. that the complement If G contains an odd circuit, but no odd hole, then it must contain a 3-circuit; but that contradicts the hypothesis o(c)= a ( G ) = 2. So G cannot contain an odd circuit, i.e., is a bipartite graph. But that is impossible since it would imply that G is a-perfect. Consequently, G must contain an odd hole; thus G contains the complement i 2 k + l ( k 2 2) of an odd circuit L2k+l as a subgraph. Since iZn+, is not perfectly a-partitionable this is possible only if G = E Z k + l . Theorem 10 is now proved. 0

c

E. Olaru, H. S a c k

138

Corollary. A graph H with w ( H )= 2 or neither H nor H contains an odd hole.

a (H) =2

is a-perfect if and only if

Proof. The assertion follows from Lemma 1 and Theorem 10. 0 Now we are able to prove the following theorem:

Theorem 11. A critically a-imperfect graph G is an odd circuit LZk+, ( k 2 2), or the complement of such a circuit, if and only if n ( G )= 2 a ( G ) + 2 w ( G ) - 3.

(20)

(See also the Remark following Corollary 2 of Lemma 4.)

Proof. (a) For an odd circuit

LZk+l

(k 3 2), clearly,

n(Lzk+i)=2k+ 1, (Y(Lzk+i)=k

and O(Lzk+i)=2,

so

L z k + l satisfies (20). If a graph G satisfies (20) then its complement too, satisfies (20), for a ( G ) =w ( G ) and w ( G ) = a ( C ) . (b) If G is a critically a-imperfect graph satisfying (20) then, by virtue of Theorem 8, n ( G ) = a ( G ) * w ( C ) + 1, therefore a (G) * o ( G ) + 1 = 2 a ( C ) 2 w ( G ) - 3 , implying w ( C ) = 2 or a ( G ) = 2 . The assertion now follows from Theorem 10. 0

c,

+

Let G = (X, U ) , Y C X;we define

r , ( Y ) = { x I x EX, x$Z Y, and there is a y E Y with ( x , y ) E U } ; we write Tc(y)instead of T C ( { y } ) .

Theorem 12. A critically a -imperfect graph G = ( X , U )or its complement G is an odd circuit L Z k + l ( k 3 2 ) if and only if for every maximum clique C of G (i.e., for every clique C = [Q], with I Q I = w ( C ) )

I rc(Q)ls w ( G ) + 1.

(21)

Proof. If the critically a-imperfect graph G or its complement circuit L 2 k + l (k 2 ) then (21) clearly holds. We shall show that (21) implies

n ( C )= 2 a ( G ) + 2 w ( G ) - 3 ; the assertion then follows from Theorem 11.

0

is an odd


139

From Lemma 4 and the Corollary of Theorem 8 we derive the following statement:

For every vertex x EX, (Y([X2]c)= a(G)-l,

i.e., w ( [ X ' ] , ) = w(G)- 1, where X 2= Tc(x). From this it follows that every vertex x of G is contained in a maximum clique, say C = C(x) = We consider two cases. (I) There is a vertex y E Q, y # x such that

[a],.

([XI U {y)]c)

= a (G) = 1,

where XI = I'c(x). [X'U{y}],, being a proper subgraph of G, property (B) of critically a imperfect graphs (see Definition 6) yields

O([X' U{y}]c)= a ( G ) - l = e ( [ X ' ] , ) . Using Lemma 4 we conclude from this relation that there exist in X' at least two vertices which in G are adjacent to y. By hypothesis,

I rc(Q)Is w(G)+

1,

implying

I rc (x)-

(Q -{x))I

I Tc (Q) I - 2

w(G) - 1,

thus

I X 2I = I T C ( XI =) I T c ( x ) - ( Q -{XI) I + I Q - { x } 1

2w(G)-2,

and because of Corollary 2 of Lemma 4 (see (***)),

I X21 = 2 w ( G ) - 2 . (11) For every vertex y E Q, we have a ([XI U { y ) ] ~ = ) a (G). Let y E Q, y # x. By Lemma 4 (see (**)), (Y

([x2- { x '}I c ) = (Y (G ) - 1

or, equivalently, ~ ( [ X ' - { X ~ } ]w(G)-1 ~)=

for every x 2 E X 2 .

This implies the existence of a maximum clique C' = [ Q']c with x E Q', y $Z Q '.

E. Olaru, H. S a c k

140

For Q’ either case (I) holds (then 1 X z I = 2 w ( G ) - 2) or case (11) holds, i.e., &([XIU {y‘}],) = a(G) for every y ’ E 0‘. Now, in Q’ - { x } there is a vertex z such that ( y , z ) fZ U for, otherwise, Q ’ U { y } would span a clique in G with w ( G ) + 1 vertices, contradicting the definition of w ( G ) . The edges ( x ,y ) and ( x , z ) with ( y , z ) U are a-critical with respect to G ; consequently, because of Theorem 6, G is an odd circuit L 2 k + , (k 3 2) and thus, clearly,

1 X 2 1 = 2 w ( G )- 2 . So, in every case, we have obtained IX21=2w(G)-2 and, analogously,

1 x’I = 2 w ( G ) - 2

=2a(G)-2.

Hence,

n ( C )= I X iI + 1 X21+ 1 = 2 w ( G ) + 2 a ( G ) - 3 , proving Theorem 12. 0

Remark. Clearly, a critically a -imperfect graph G is a-non-partitionable (see proof of Theorem 7); this, in connection with Theorem 1, implies: (i) G is connected, (ii) no separating vertex set of G can span a clique in G . Theorem 13. Let T be a separating vertex set of the critically a-imperfect graph C = ( X , U ) such that a ( C - T )= a ( C ) . Then T spans a connected subgraph in the complement G of G . Proof. Assume that [TIt: is disconnected. Then T can be partitioned into two parts A and B such that A#0,

B#O,

A n B = 0 , A U B = T and ApcB.

Since, by hypothesis, C - T is disconnected, it can be decomposed into two subgraphs GI = [ X , ] , and G2= [X2IGsuch that

X , # 0, X 2# 0, X , f l X 2 = 0, X , U X , = X - T and

X,P,X,.


141

B # 0 implies that [XIU X2 U AIc is a proper subgraph of G ;therefore, because of property (B) of G (see Definition 6), 8 ([Xi U Xz U A ]G) =

([Xi U X2 U A ] G ) = ( G ) .

(23)

Let denote a perfect a-partition of [Xi U X , U A]G and let, for i = 1,2, A, be the set of all those vertices of A contained in parts (ie., cliques) of the partition Z which also contain at least one vertex of X,. Then A , n A, = 0 and A 1U A, = A since two vertices x1 E XI and x2 E Xzcannot be contained in the same clique, and in Z there is no clique all of whose vertices belong to A (because of (22) and (23)). It is easily seen that

a ( [ X UA,],)=a(G,)

(i =1,2).

(24)

By means of a perfect a -partition of [XIU XzU BIG we define in an analogous manner the sets B1 and B2 satisfying:

B~n B , = 0,

B, u

= B,

a ( [ X UB,].)=a(G,) (i=l,2).

(25)

Further, we prove

a([Xi U A i U B , ] G ) = a ( G i ) (i =1,2). Let S be a stability system of [XlU A, U B1IG.If S C X , then, clearly, (26) is true (for i = 1); if S n A , # 0, then S n B , = 0 since AlpcBl (because of ApGB); analogously, if S n B , # 0, then S n A , = 0. The subscript 1 can be replaced by 2, and we conclude from (24) and (25) that (26) is valid in every case. Because of property (B) of G (see Definition 6), we obtain from (26) and (22): 8 ( G )= O([XiU X2U A U B I G )

e([x,u A, u B , I ~ ) +e([x2u A, u ~ = a([Xi U A I U Bi]c)+

(~([xz U A2U

~

1

~

)

&]G)

+

= a ( G , ) a(G2) = ( Y ( G ) ,

contradicting property (A) (see Definition 6) of critically Theorem 13 is now proved. 17

(Y

-imperfect graphs.

Theorem 13 has some important consequences. As a first conclusion, we obtain the following corollary: Corollary 1. Let G be a critically a-imperfect graph and let [Q], be an arbitrary clique of G. Then G - Q is connected.

142

E. Olaru, H.Sack

Proof. We may assume the clique [QIG to be maximal, so I Q 13 2. We have a(G - Q ) = a ( G ) .The graph [ Q ] B consists of isolated vertices only, hence, by Theorem 13, Q cannot separate G. 0

Remark. Corollary 1 of Theorem 13 can be used to immediately prove the following well-known theorem (see also [13], Theorem 1 (p. 150) and Folgerung aus Satz 4 (p. 153)): Theorem (Hajnal and Suriinyi [9], Berge [4]). Every triangulated graph is perfect. Let X' denote a separating vertex set of G = ( X , U ) and let C = [ YIGbe a connected component of [ X - XrlG. If in G every vertex of X' is adjacent to at least one vertex of Y then C is called a normal component with respect to X ' ; if [ X - X'Ic has at least two components which are normal with respect to X'then X' is called a normal separating vertex set of G. Corollary 2. Let G = ( X , U ) be a critically a-imperfect graph and let x be an arbitrary vertex of G. Then (a) the graph Glxl= [ X ' ] , is connected, and (b) XI is a normal separating vertex set of G.

Proof. (a) Clearly, for every x E X the vertex set X ' = r c ( x ) is a separating vertex set of the critically a-imperfect graph G. Further, a(G - X ' ) = a ( [ X 2U { x } ] ~ = ) a ( G ) since every vertex of the critically a-imperfect graph is contained in at least a ( G )stability systeps of (see Corollary to Lemma 3). The assertion now follows from Theorem 13. (b) Proposition (a) applied to G says that [ X ' ] , is connected: thus X' separates G, and G -X'has precisely two connected components, namely, [X2]a and the isolated vertex x. Since, by definition, xpc;-X'all we have to show is that in C every vertex of XI is adjacent to some vertex of X 2 . Assume that there is a vertex x ' E X ' which, in G, is not adjacent to any vertex of X2.Then, in G, x is adjacent to every neighbour of X I , i.e., the graph GixlJ= [Tc(x')lCis disconnected, contradicting (a). 0

c

It is convenient to restate Corollary 2 of Theorem 13 in the following equivalent form.

Corollary 2'. For every vertex x of a critically a-imperfect graph G the following statements hold : (a') the graph Gixl is connected, (b') the set X z of the neighbours of x is a normal separating vertex set of G.


143

Remark 1. Corollaries 1 and 2 of Theorem 13 can be proved without using LovBsz’ theorem (see Theorem 5). Remark 2. Proposition (a) of Corollary 2 of Theorem 13 can be used to prove Dilworth’s theorem (see [13], Th. 3 (p. 150 and p. 158)): Theorem (Dilworth [S]). Every transitively orientable graph (‘comparability graph’) is a -perfect. Proposition (b) of the same corollary yields a simple proof of Gallai’s theorem (see [13], Th. 2 (p. 150 and pp. 164-165)):

Theorem (Gallai [7]). Every graph G having the property that each of its odd circuits (of length 2 5 ) is triangulated by some of its chords is a-perfect. Concluding Remark. The strong version of the Perfect Graph Conjecture (see Conjecture 1) is equivalent to the following: Conjecture 1’. The only critically a-imperfect graphs are the odd circuits of length 3 5 and their complements. By virtue of Theorem 10, Conjecture 1’ can also be given the following equivalent formulation:

Conjecture 1”. The only critically a-imperfect graphs G with a(G)> 3 are the odd circuits of length 2 7 . Therefore, it remains a central task to investigate the structure of critically a -imperfect graphs. In this direction, a somewhat weaker result obtained by Olaru in 1969 (see [13], Satz 14 and Folgerung (p. 169) or [17], Satz 3.1 (p. 99)) should be mentioned:

An a-imperfect graph G = ( X , V ) which has the property that G - u is a-perfect for every edge u E U is called edge-critically a-imperfect. Clearly, an edge-critically a -imperfect graph without isolated vertices is (vertex-) critically a -imperfect. Theorem 14. Every edge-critically a-imperfect graph consists of an odd circuit of length 2 5 and, possibly, some additional isolated vertices.

E. Oluru. H. Suchs

144

References [ I ] C. Bergc, Graphes et Hypergraphcs (Dunod, Paris, 1970). 121 C. Berge, Firbung von Graphen deren s8mtliche bzw. deren ungerade Kreise starr sind (Zusamrnenfassung). Wiss. Z . Martin-Luther-Univ. Halle 10 (1961) 114-1 15. [3] C. Berge, Une application de la theorie des graphes a un problkme de codage, in: Caianello. ed.. Automata Theory (Academic Press. New York-London. 1966) 25-34, [4] C. Berge, Some classes of perfect graphs, in: Graph Theory and Theoretical Physics, Chapter 5 (Academic Press, London-New York, 1967) 355-166. [ S ] R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. Math. 51 (1950) 161. [h] D.R. Fulkerson. Anti-blocking polyhedra, J. Comb. Theory 12 (1972) 50-71. 171 T. Gallai. Graphen init triangulicrbaren ungeraden Vielecken, Magyar Tud. Akad. Mat. Kutat6 Int. K(izl. 7A (1962) 3-37. [ X I P.C. Gilmore and A.J. Hoffman, A characterization o f comparability graphs and of interval graphs, Canad. J . Math. I6 (I‘J64) 53Y. [ 9 ] A. Hajnal and J. Suranyi. IJber die Aufltisung von Graphen in vollstandige Teilgraphen, Ann. Univ. Sci. Budapest I (19%) 1 1 3 . [lo] L. Lovasz. A characterization of perfect graphs, J. Comb. Theory 13 (1972) 95-98, [ I ] ] L. Lovasz, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-267 (this volume, pp. 29-42). (121 S.E. Markosjan and I.A. Karapetjan, Perfect graphs (in Russian), Akad. Nauk Armjan. SSR Dokl. 63 (1976) 292-296. [ 131 E. Olaru, Beitrage m r Thcorie der perfekten Graphen, Elektronische Informationsverarbeitung und Kybernetik (EIK) S (1972) 147-172. [14] E. Olaru. Zur Charakterisierung perfekter Graphen, EIK 9 (1973) 543-548. [IS] E . Olaru, Uber perfekte und kritisch imperfekte Graphen, Ann. St. Univ. Iagi 9 (1973) 477-486. [I61 E. Olaru. Uber die Uberdeckung von Graphen init Cliquen, Wiss. Z. TH Ilmenau 15 (1969) 115-120. 117) E. Olaru. Zur Theorie dcr perfekten Graphen, J . Comb. Theory, Ser. B 23 (1977) 94-105. [IS] H. Sachs, On the Berge conjecture concerning perfect graphs, in: Cornbinatorial Structures and Their Applications (Gordon and Breach. New York. 1970) 377-384. [ 191 W. Wessel, Some color-critical equivalents of the strong perfect graph conjecture, in: Beitrage

zur Graphentheorie und deren Anwendungen (Proc. Int. Koll. “Graphen theorie und deren Anwendungen”. Oberhof (DDR), I ( k l 6 April 1Y77) (Math. Gcs. D D R , TH Ilmenau, 1977)

30(k309.


MEYNIEL’S GRAPHS ARE STRONGLY PERFECT G. RAVINDRA* ER 175 Combinatoire, Uniuersite‘ Pierre el Marie Curie, UER 48, 75230 Paris, France Meyniel (see article this volume, pp. 115-119) proved that a graph is perfect whenever every odd cycle of length at least five has at least two chords. This paper strengthens this result by proving that every graph satisfying Meyniel’s condition is strongly perfect (i.e., each of its induced subgraphs H contains a stable set which meets all the maximal cliques in H ) .

The graph-theoretic notions used here are those of [3]. A graph is called perfect if the chromatic number of each of its induced subgraphs H equals the number of vertices in the largest clique of H ; it is called strongly perfect [4] if each of its induced subgraphs H contains a stable set which meets all the maximal cliques in H. (Here, as usual, ‘maximal’ is meant with respect to set-inclusion.) It is easy to show that every strongly perfect graph is perfect. Meyniel [5] proved that a graph is perfect whenever each of its odd cycles of length at least five has at least two chords. The purpose of this paper is to strengthen Meyniel’s result as follows: Theorem. If every odd cycle of length at least ,five in a graph G has at least two chords then G is strongly perfect.

The following useful observation has been made by Meyniel. Lemma 1. If a graph G = ( V ,E ) contains an odd cycle [xo, xl,. . . , xzr,xd] such that the path [x,, xz, . . .,x Z f ]is chordless and xo is nonadjacent to at least one x k , then G contains an odd cycle of length at least five with at most one chord.

Proof. If x o x z E E then consider the largest i such that xo is adjacent to x1x2.- ax, and the smallest j such that j > i and xoxJ E E : the cycle [xo, x,, . . . ,x,, xo] has no chords, the cycle [xo,x , - ~ ,x,, . . . ,x,] has precisely one chord, and one of these two cycles is odd. If xoxz! i ! E then consider the smallest even j such that xoxJ E E and the largest i such that i < j - 2 and xox, E E : the odd cycle [xo,x,, ..., x,,xo] has at most one chord. 0 * Current address: Regional College of Education, Ajmer 305 004, India. 145

G. Ravindra

146

By a starter in a graph G, we shall mean a cycle wu0ul * . . U k W such that (i) uo is adjacent to none of the vertices u2, u 3 , . . . ,uk, (ii) w is not adjacent to uI, (iii) some stable set S, containing uI and u k , meets all the maximal cliques in

G - uO.

Lemma 2. lf n graph contains a starter then it contains an odd cycle of length at least five with at most one chord.

Proof. We shall present an informal description of an efficient algorithm which, given any starter in G = (V, E), finds the desired cycle. No generality is lost by assuming that (iv) [ ul, u z , . . . ,u k , w ] is the shortest path from u1 to w with the next-to-last vertex in S and all the vertices except the first and the last nonadjacent to uo. In particular, (iv) implies that (v) the path [ul, u 2 , .. .,uk] has no chords. Next, we may assume that (vi) every u, adjacent to w has an even subscript r ; otherwise the desired cycle can be found at once by applying Lemma 1 to the odd cycle W U o * . U,W. Write y E S" if y E S and y is adjacent to two consecutive vertices u,, u , + ~on the path [uO,u I , .. . , u k ] . We may assume that (vii) no y E S" is adjacent to uo; otherwise the desired cycle can be found at once by applying Lemma 1 to any odd cycle [ y u o - . u,y]. Now it follows that (viii) no y E S" is adjacent to w ; otherwise (iv) would be contradicted by uoul * * . u,yw such that i is the smallest subscript with yo, E E. Next, we may assume that (ix) each y E S * is adjacent t o at least three vertices on the path u0ul Uk ; otherwise the desired cycle is [ w , u,, . .. ,u,, y , u , + ~.,. . , u,, w ] with r standing for the largest subscript such that r < j, wu, E E and s standing for the smallest subscript such that s 3 j + 1, wu, E E. Now it follows that u,, u,+] on the path (x) each y E S " is adjacent to precisely three vertices [ U h 01,. . . , u t ] ; otherwise (iv) would be contradicted by uI . u,yu, - Dk such that r is the smallest subscript with yu, E E and s is the largest subscript with yu, E E. u, and u , + ~the , substitution Now observe that, for any y E S* adjacent to of y for u, in the original starter yields a new starter with a smaller S * .Repeating this operation as many times as possible, we eventually obtain a starter wVo& * Vkw satisfying (iv) and having an empty S*. Since k is even by (vi) and since V,, 6k E S, there is an edge fi,Gf+l with j > 0 and neither endpoint in S (it

-

4

4

Meyniel's graphs are strongly perfect

147

suffices to set j = i - 2 with i being the smallest even subscript such that 17, E S ) . Let C be any maximal clique extending 6,17,+~in G - uo. Since S * is empty, C is disjoint from S, a contradiction. 0

Proof of the Theorem. We shall present an informal description of an efficient algorithm which, given any graph G, finds either a starter in some induced subgraph of G or else a stable set meetirlg all the maximal cliques in G. The set of neighbours of a vertex u in G will be denoted by N ( u ) . First, choose a vertex t and a component H of G - t - N ( t ) so that the number of vertices in H is minimized (over all choices of t and H ) . If H = 0 then set S = { t } ;otherwise choose a vertex uo in H and denote by F that component of G - uo - N(uo) which contains t. (For future reference, note that each vertex x in H - uo - N ( u o ) belongs to F : otherwise the component of G - uo - N(uo)containing x would be fully contained in H - uo, and so it would have fewer vertices than H.) Apply the algorithm recursively to G - uo. When a stable set S meeting all the maximal cliques in G - vo is returned, search for vertices u l , u2 in G - uo such that uI E N(u o )fl S

and

v2 E N(vl)fl F.

If such vertices cannot be found then the stable set ( S - N(uo))U {uo}meets all the maximal cliques in G. (To verify this claim, assume that some maximal clique Q in G is disjoint from ( S - N(uo))U {uo}. Since uoE? Q, there is a vertex x E Q - N(uo) and there is a vertex uI E Q fl S. Of course, u1 E N(uo). If u1 E N ( t ) then we may set u2 = t ; otherwise u1E H and so x E H U N ( t ) , in which case we may set u2 = x.) If uI and u2 are present, search for vertices w and z , distinct from uo, u l , u2 and each other, such that w E N(u o )- N ( u l ) and

z E N ( w )fl F f S. l

If such vertices cannot be found then S meets all the maximal cliques in G. (To verify this claim, assume that some maximal clique Q in G is disjoint from S. Since u1E S, at least one vertex w E Q is nonadjacent to u I . Note that uo E Q, and so either (Q - vo) fl H # P, or else Q - v o c N ( t ) . Let Q " be any maximal clique in G - uo which extends Q - uo in the first case and (Q - uo)U { t } in the second case. In either case, we have Q * H U N ( t ) U { t } , and so Q * - N(uo) F. The vertex z common to Q * and S must be outside N(uo):otherwise Q could have been extended to Q U {z}.) If w and z are present then any path from uz to z in F yields a starter with v k = z. 0 Acknowledgements

The author thanks Prof. V. Chvatal for simplifying the proof of the main theorem and rendering the presentation more lucid. The author is also grateful to Prof. Claude Berge for fruitful discussions.

148

G. Ravindra

References [ 11 C. Berge, Farbung von Graphen, deren samtliche bzw. deren ungerede Kreise starr sind, Wiss. 2. Martin-Luther-Univ. Halle-Wittenberg 114 (1961). [2] C. Berge, Sur une conjecture relative au probleme des codes optimaux. comm. t 3 h e assemblee gtnerale de I’URSI, Tokyo (1962). [3] C. Eierge, Graphs and Hypergaphs (North-Holland, Amsterdam, 1973). (41 C. Berge and P. Duchet, Strongly perfect graphs (this volume, pp. 57-61). [ 5 ] H. Meyniel, On the Perfect Graph Conjecture, Discrete Math. 16 (1976) 339-342.


THE VALIDITY OF THE PERFECT GRAPH CONJECTURE FOR K,-FREE GRAPHS Alan TUCKER Department of Applied Mathematics, State University of New York. Stony Brook, N Y 11 794, USA

1. Introduction

This paper builds on results based on D.R. Fulkerson's antiblocking polyhedra approach to perfect graphs to obtain information about critical perfect graphs and related clique-generated graphs. Then we prove that the Perfect Graph Conjecture is valid for 3-chromatic graphs. Fulkerson felt that a proof of the Perfect Graph Theorem would involve exactly the kind of duality that existed in his theory of blocking and antiblocking polyhedra [S], [6]. He proved what he called the Pluperfect Graph Theorem [7]. Lov6sz [8]independently developed some of the antiblocking theory, stated in hypergraph terms, and showed that with a fairly simple lemma, the Pluperfect Graph Theorem implied the Perfect Graph Theorem. Lovkz [Y] obtained a related result (Theorem 1 below) with a valuable implication about critical perfect graphs. A critical perfect graph, p-critical for short, is an imperfect graph all of whose proper induced subgraphs are perfect. The major results about p-critical graphs, especially those of Padberg [lo], are based o n an antiblocking polyhedra approach (also see [ 3 ] ) .

Theorem 1 [Y]. A graph G with n vertices is perfect if and only if a ( C ) w ( G ) for all induced subgraphs G' of G. Thus a p-critical graph G has ti a ( G ) w ( G ) + 1 vertices.

11.

=

Theorem 2 [lo]. A p-critical graph with n uertices has exactly 11 cliques of size w ( G ) with each vertex in w ( G ) maximal cliques and has exactly ri stable sets of size a ( G ) with each vertex in a ( G ) maximal stable sets. Each maximal clique intersects all but one maximal stable sets, and vice versa. We define a graph G to be pseudo-p-critical if (a) G has n = a ( G ) w ( G ) +1 vertices; 149

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A. Tucker

(b) for all x in G, a (G - x ) = B(G - x ) and w ( G - x ) = y(G - x ) , (c) G has the properties listed in Theorem 2. Clearly a p-critical graph is pseudo-p-critical.

2. Unique covers and connectivity in G - x

Throughout this section, we assume that G = (V, A ) is a pseudo-p-critical graph with a = a ( G ) , w = w ( G ) , and n = I V ( = a w + 1. Then for any x E V, G - x is covered by a cliques of size o and also by w stable sets of size a (by property (b)). If C is one of the a cliques in the cover of G - x , then a ( G - x - C )= a - 1 and so if Z is the maximal stable set such that I f l C = 0, then x E Z. On the other hand, for each x E Z, one clique of the a clique cover of G - x will not intersect I; so C must be in all those clique covers. Further, C cannot be in any other clique cover of G - y for Z, or else C would have to intersect f just as it intersects all maximal independent sets in G - y. A similar argument applies to Z's appearances in stable set covers of G - x. So we have proved the following theorem:

ye

Theorem 3. I f G is pseudo-p-critical and I and C are an a-stable set and an w-clique, respectively, of G with Z f l C = 0, then C is in an a clique cover of G - x if and only if x E Z, and Z is in an w stable set cover of G - x if and only if x E C. CoroUary 3.1. Let G be pseudo-p-critical and x be any vertex in G. Then the w stable set cover (w-coloring) and the a clique cover of G - x are unique. Corollary 3.2. Let G be pseudo-p-critical, x be any vertex in G,and I,,Zz be two color classes in a w-coloring of G - x. Then the subgraph induced by Z, f l I2 is connected.

Proof. If not connected, the color classes could be interchanged in one component to get a different coloring, violating Corollary 3.1. 0 Corollary 3.3. Let G be pseudo-p-critical and C be a maximal clique of G. Then as y ranges over the w different vertices in C,the clique covers of the G - y contain all the other n - 1 ( = a w ) maximal cliques.

Proof. If the same clique was in covers for different y's, then by Theorem 3 the y's must be in a (maximal) stable set. 17 Let N ( x ) denote the set of x and the vertices adjacent to x.

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Theorem 4. Let G be pseudo-p-critical and x be any vertex in G. Then G - N ( x ) is connected.

Proof. Suppose G - N ( x ) is not connected. Let GI be one component of G - N ( x ) and let Gz be the rest of G - N ( x ) . Let C1be a clique of the a clique cover of G - x and which is contained in GI U N ( x ) . Let Czbe defined similarly for GzU N ( x ) . Let a, = a ( G i ) , for i = 1,2. Clearly, a = (G - N ( x ) ) + 1 = a l + a Z + 1 ,and a ( G - C , ) = a , for i = 1 , 2 . This implies a ( G , - C , ) = a , , for i = 1,2. But then a -1 = a ( G - C1- Cz)= a ( ( G- N ( x ) ) - C1- G ) + 1 = a (GI- C , )+ a (GZ- Cz)+ 1 = a 1+ a z + 1 = a, a contradiction. 0

3. Maximal clique graphs Let us define M ( G ) , the maximal clique graph of the graph G, to have one vertex for each maximal clique of G (of size w (G)) and an edge between vertices which correspond to intersecting cliques. First we prove a maximal clique duality for pseudo-p-critical graphs. Then we prove that M ( G ) is pseudo-p-critical if G is pseudo-p-critical. Theorem 5. If G is a pseudo-p-critical graph, each maximal clique in M ( G ) corresponds to a vertex of G.

Proof. Since by Theorem 2, each vertex x of G is in w maximal cliques, then the corresponding w vertices in M ( G ) form a clique of size w. Now suppose that there exists a set of m, m 2 w, pairwise intersecting maximal cliques in G with no common vertex. Let C,, C,, . . . , C,,, be this set of cliques and I,,l 2 ,... , I,,, be the associated maximal stable sets with Cin Ii = 0. For any two ck,C, there is a vertex y E c k n C, and then by Theorem 3 the associated I,, I, are both in the stable set cover of G - y and so Ik f l I, = 0. Since this is true for any two of the I’s, the rn I’s are mutually disjoint. Since m 2 w, then it must be that m = w. Further, these 1 ’ s must contain all but one vertex of G, call it z , and so form a stable set cover of G - z. Then by Theorem 3, each Ck contains z. 0 Theorem 6. If G is a pseudo-p-critical graph with n = a w + 1 vertices, then M ( G ) is also pseudo-p-critical with n vertices and w ( M ( G ) )= w and a ( M (G)) = a.

Proof. By Theorems 2 and 5, we know that M ( G ) has n vertices, that w ( M ( G ) )= w, and that M ( G ) has exactly n cliques of size w. Since for any x, G - x has a cover of a (disjoint) maximal cliques, it follows that a ( M ( G ) )= a.

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Conversely, any stable set of a disjoint maximal cliques is a cover of G - z, for some z, and so by Corollary 3.1 M ( G ) has exactly n maximal stable sets. Clearly each maximal clique contains w vertices and by Theorem 3, is in a clique covers and so each vertex of M ( G ) is in w maximal cliques and a maximal stable sets. Also the maximal clique of M ( G )corresponding to the set of maximal cliques of G containing the vertex x is uniquely disjoint from the maximal stable set of M ( G ) corresponding to the a cliques covering G - x. Thus M ( G ) satisfies the properties of Theorem 2. If C is any maximal clique of G and I is the associated disjoint maximal stable set, then every other maximal clique of G contains one of the vertices in I and so the a cliques in M ( G ) corresponding to the vertices in I constitute a clique cover of M ( G ) - C of size a. Thus B ( M ( G ) - C ) = a. O n the other hand, by Corollary 3.3, the w stable sets of M ( G ) , corresponding to the clique covers of G - x for the w different x’s in the maximal clique C, cover M ( G ) - C. So y ( M ( G ) )- C ) = w. Corollary 6.1. If M ( G ) is the maximal clique graph of a pseudo-p-critical graph G,then for a vertex C in M ( G ) , M ( G ) - N ( C ) is connected.

Proof. This is an immediate consequence of Theorems 4 and 6. 0 For a given graph G, defined the skeleton S ( G )to be the subgraph of G with the same vertex set as G and only those edges of G that are part of a maximal clique of G.The following theorem follows from Theorems 5 and 6. Theorem 7. If G is pseudo-p-critical, then M ( M ( G ) )= S ( G ) and so S ( G ) is also pseudo -p-critical.

4. Perfect 3-chromatic graphs In this section, we prove that Berge’s Perfect Graph Conjecture is true for 3-chromatic graphs. First we prove a lemma that is of some interest in its own right. Lemma 8. Let G be pseudo-critical and let x be a vertex of G. Then, in the w -coloring of G - x , each i - j edge is on an i - j path whose terminal vertices are in N ( x ) . Proof. Let F stand for the subgraph of G induced by x along with all the i-vertices and all the j-vertices. If the lemma is false then, by Menger’s theorem,

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153

there is a vertex s such that x and some i - j edge uv belong to distinct components of F - s. Let H stand for that component of F - s which contains the edge uu. We may assume that s is an i-vertex. Now consider the cover of G - s by CK cliques. Each of these cliques except one includes one vertex of each color; the exceptional clique includes x and one vertex of each color distinct from i. Thus for each i-vertex w other than s there is a unique j-vertex f ( w ) such that w and f ( w ) belong to the same clique in the cover. Since f is one-to-one and since f ( w ) E H if and only if w E H, it follows that the number of i-vertices in H equals the number of j-vertices in H. Choose a j-vertex t in H and consider the cover of G - t by CK cliques. Each of these cliques except one includes one vertex of each color; the exceptional clique includes x and one vertex of each color distinct from j. Thus for each i-vertex w in H there is a unique j-vertex g ( w ) in H such that w and g ( w ) belong to the same clique in the cover. Since g is one-to-one, it follows that the number of i-vertices in H is at most the number of j-vertices in H - r, a contradiction. 0

Theorem 9. Every graph containing no holes, no anthiholes, and no K4 is perfect. Proof. Assume the theorem false and let G be a counterexample with n vertices and rn edges such that no counterexample has fewer than n vertices and no counterexample with n vertices has fewer than rn edges. Since n is minimal, G is p-critical; since rn is minimal, every edge of G belongs to a triangle. (To clarify the second point, assume that an edge e of G belongs to no triangle. Since G - e retains the n triangles covering each vertex three times, we have CK (G - e ) s n / 3 = a ( G ) + f, and so G - e is imperfect. Furthermore, G - e has no holes, for otherwise G would have a hole. Finally, G - e has no antiholes of length at least seven, for otherwise G - e would have a K4. Thus G - e is another counterexample, a contradiction.) By Theorem 2, each N ( x ) in G induces three triangles. If these three triangles are arranged as in Fig. 1 for every x, then an easy argument (whose details we omit) shows that G is completely determined: its vertices can be enumerated as u , , v z , . . . ,u. ( n = 3 a ( G ) 1 ) in such a way that, with subscript arithmetic modulo n, vertices v, and v, are adj icent if and only if I i - j 1 s 2. But then G is either an antihole (if a ( G ) = 2) or tqsily seen to contain a hole (if a ( G ) 2 3). (Generalizations of the latter case are given in Tucker [13] and Chv5tal [4].) Now choose a vertex x such that the three triangles xcr', xdr and xab are not arranged as in Fig. 1 and consider the unique 3-coloring f of G - x. By the second part of Theorem 3, we may assume that

+

f ( c ) = 2, f ( r ' ) = 3 and f ( d ) = 1 , f ( r ) = 3 and f ( a ) = 1 , f ( b ) = 2 .

A. Tucker

154

X

Fig. 1.

Thus w e may assume that N ( x ) is arranged as in Fig. 2, possibly with r = r ' . If e is an edge on a 1-2 path w I ,w 2 , . .., wk with w I , wk E N ( x ) and w 2 ,w,, . . . ,Wk-I fZ N ( x ) then we shall write if w I = a and

Wk

= d,

e E S 2 if w I = b and

wk

= c,

e E S 3 if w I = a and

wk

=

e E SI

b

b2 2c r'3

3s & r2

Fig. 2.

(see Fig. 2). We claim that every 1-2 edge in G - x belongs to precisely one of the three sets S. To justify this claim, we first observe that every 1-2 path W I , W 2 , . . . , Wk with W l , Wk E N(X) and W 2 , W 3 , . . . , Wk-1 p N(X) must have { w l ,wk} = { a , b } , { b , c } or {a,b ) : otherwise w I , wk would be nonadjacent and have distinct colors, in which case the shortest path from w I to wk through w2, w 3 , ..., wk-1 would, together with x, yield a hole. This observation alone implies that SI,S2, S3 are pairwise disjoint; combined with Lemma 8, it shows that every 1-2 edge belongs to exactly one S,. In addition, note that Sl and Sz are nonempty since (by Corollary 3.2) the 1-2 subgraph of G - x is connected. Corollary 6.1 with the triangle abx in place of C guarantees that the hypergraph consisting of all the triangles disjoint from abx is connected. In particular, some triangle disjoint from abx and with a 1-2 edge in S1 shares its 3-vertex z with a triangle disjoint from abx whose 1-2 edge does not belong to S , . For simplicity of exposition, let us first assume that z#r

and

zZr'.

Let the three triangles which contain z be named C1,C2,C3 in such a way that one of the following five statements is true.

The validity of the Perfect Graph Conjecture for &free graphs

155

(i) each C, has its 1-2 edge in a different S,. (ii) C1has its 1-2 edge in SI and Cz, C3 have their 1-2 edges in S3, (iii) C1has its 1-2 edge in S1 and Cz, C3 have their 1-2 edges in Sz, (iv) C1has its 1-2 edge in S1 and Cz, C3 have their 1-2 edges in S1, (v) C1has its 1-2 edge in S3 and Cz, C3 have their 1-2 edges in S1. In case (i) or (ii), let vI, vz,. . . ,us be a shortest path with v I = z , v z = a and v2v3, v3v4,. . . ,vo-lvs E S3, and v,# b for all i. Let w l , w z , .. ., w, be a shortest path with w, = a, wl = d and w 1 w 2 w 3 ., ..,wl-lwl E S1 and wlwl+lE C1for some k. Now consider the closed walks V I V Z " ' V ~ W ~ " 'W k V i

and

V r V 2 " ' V S X W ~ W ~ - ~ " 'W k + I V I

of lengths s + k - 1 and s + 1 + t - k, respectively. Since t is odd (each w, is colored 1 if j is odd and 2 if j is even), precisely one of the two walks is odd. We claim that this odd walk L is a hole. To justify this claim, consider first the case when the odd walk is

L

=

vIv2"'

Vsw2. ' . Wkvl.

Since s + k - 1 is odd and s 2 2, k 2 1, we have s + k - 1 2 3. In fact, we cannot have s + k - 1 = 3, for then either s = 2, k = 2 and z is in two triangles with 1-2 edges in Sl, or else s = 3, k = 1 and s is not minimal. Furthermore, L has no chords v,v, (by minimality of s), no chords w,w, (by minimality of t ) , no chords v,w, (since z is in only one triangle whose 1-2 edge belongs to S , ) and no chords v,v,, 1 < i < s (for such chords would belong to both Sl and S3).Thus L is a hole. Next consider the case when the odd walk is

L

=vIvZ"'~sxwiwi-I"'Wk+lvI.

Since s + 1 + t - k is odd and s 2 2 , t - k + 1 3 3 , we have s + 1 + t - k 3 5 . As before L has no chords QV,, no chords w,w, and no chords v,w,, i < s. There are no chords beginning at x since N ( x ) = {a, b, c, d, r, r'}. To prove that there are no chords v,w, (i.e., aw,), recall that no vertex of G has four pairwise nonadjacent neighbors. Since x , us-, and wz are pairwise nonadjacent neighbors of v,, then or wz. The first endpoint w, of chord v,w, would have to be adjacent to x, two options are easily eliminated; the third is unavailable since the 1- or 2-vertex w, cannot be adjacent to both the 1-vertex v, and the 2-vertex w2. Thus L is a hole. In case (iii), let vl, vz,. . . , v, be a shortest path with vr = z and v, = x , and V Z v 3 , v3v4,.. . ,U , - ~ U , - ~E S2 and v,# b for all i. Let w,, w z , .. . , wl be as before. By the same reasoning used above, one can verify that one of the closed walks ViVz'

is a hole.

' *

V,Wi

* *

WkVi

and

ViVz'

.

'

V,WrWl-r

* * .

Wk+IVI

A. Tucker

156

Case (iv) reduces to (iii) by symmetry. In case (v), let ul, u 2 , . . . , us be a shortest path with u I = z, us = a and u203,. . . , E SI. Let wI,w 2 , .. . ,wI be a shortest path with w1 = a, w, = b and w ,w 2 , .. . , wl-l wIE S,. Note that t is even since a, b have distinct colors and that 1 < k < t - 1 since C1is disjoint from abx. Now one of the two closed walks UlU2"'

U,WI"'

WklJl

and

WIU~"'U,WIWI-I"'

Wk+iUi

is odd. Again this odd walk will be a hole. Finally, if z = r or z = r', then one of the z's triangles contains x, another, which we will call CI,has its 1-2 edge in SIand the third C2has its 1-2 edge in S2 or S,. If C2has its 1-2 edge in S3 then we may proceed as in case (v). If Czhas its 1-2 edge in S2 then we may assume z = r' (the case z = r reduces to z = r' by symmetry; it is also possible that z = r = r') and proceed as in case (iii) with r'x in place of u 1 u 2 .. . us. 0 Corollary 9.1. Euery planar graph containing no holes is perfect.

Proof. It will suffice to prove that every planar graph G containing no holes is w(G)-colorable. We shall prove this by induction on the number of vertices in G. If o(G)s 3 then the conclusion is immediate: since no planar graph contains an antihole of length more than five, G satisfies the hypothesis of Theorem 9. If w ( G ) = 4 then G is easily seen to be the union of graphs G,and GZ, each of them having fewer vertices than G, such that GI n G2 is a triangle. By the induction hypothesis, GI and G2 are four-colorable; now G is easily seen to be four-colorable.

0

References [ I ] C. Berge, Farbung von Graphen, deren samtliche bnv. deren ungerade Kreise starr sind, Wiss. 2. Martin-Luther-Univ. Halle-Wittenberg Math.-Natut. Reihe 114 (1961). 121 C. Berge, Introduction a la theorie des hypergraphes, Lecture Notes, UniversitC de Montreal (Summer 1971). [ 3 ] V. Chvatal, On certain polytopes associated with graphs, J. Comb. Theory, Ser. B 18 (1975) 138- 154. [4] V. Chvatal. On the strong perfect graph conjecture, J. Comb. Theory, Ser. B 20 (1976) 139-141. [S] B.R. Fuikerson, Blocking and anti-blocking pairs of polyhedra, Math. Program. 1 (1971) 168- 194. [ 6 ] D.R. Fulkerson. Anti-blocking polyhedra, J. Comb. Theory 12 (1972) 50-71. 171 D.R. Fulkerson, On the perfect graph theorem, in: T.C. Hu and S. Robinson, eds. Mathematical Programming (Academic Press, New York, 1973) 6%76. 181 L. Lovasz, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 25F267 (this volume, pp. 29-42). "41 L. Lovasii, A characterization of perfect graphs, J. Comb. Theory, Ser. B 13 (1972) 95-98.

The validity of the Perfect Graph Conjecture for K4-free graphs

151

[lo] M. Padberg, Perfect zero-one matrices, Math. Program. 6 (1974) 18CL196. [ll] K.R. Parthasarathy and G . Ravindra, The strong perfect graph conjecture is true for K,,-free graphs, J. Comb. Theory, Ser. B 22 (1976) 212-223. [12] A . Tucker, The strong perfect graph conjecture for planar graphs, Canad. J. Math. 25 (1973) 103-114. [13] A . Tucker, Coloring a family of circular arcs, SIAM J. Appl. Math. 29 (1975) 493-502.


PART 111

POLYHEDRAL POINT OF VIEW



THE STRONG PERFECT GRAPH THEOREM FOR A CLASS OF PARTITIONABLE GRAPHS Rick GILES* School of Computer Science, Acadia University, Wolfville, NS, Canada

L.E. TROTTER, Jr.** School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY 14853, USA

Alan TUCKER*** Department of Applied Mathematics, State University of New York, Stony Brook, NY 1 1 794, USA A simple adjacency criterion is presented which, when satisfied, implies that a minimal imperfect graph is an odd hole or an odd antihole. For certain classes of graphs, including K,,,-free graphs, it is straightforward to validate this criterion and thus establish the Strong Perfect Graph Theorem for such graphs.

If graph G is imperfect but contains no imperfect proper induced subgraph, then G is minimal imperfect. Examples of such graphs are chordless odd cycles of length at least five and their complements, respectively termed odd holes and odd antiholes; Berge has conjectured that these are the only examples of minimal imperfect graphs. Conjecture 1 (Strong Perfect Graph Conjecture [l]). G is a minimal imperfect graph if and only if G is an odd hole or an odd antihole.

Whereas the reverse implication (‘if’) here is clearly valid, the forward implication (‘only if’) has been verified only for special classes of graphs [2,8, 12, 13, 15, 16, 17, 18, 191 and its general form has remained unsettled for some 20 years. We shall assume henceforth that G is a minimal imperfect graph. A direct approach to establishing Conjecture 1 leads naturally to the consideration of two types of information: properties of G which are a direct consequence of minimal imperfection and sufficient conditions for a minimal imperfect graph to be an * Partially supported by NSF Grant MCS8002987. ** Partially supported by NSF Grant ENG7809882. * * * Partially supported by NSF Grant MCS7904489. 161

R. Giles el al.

162

odd hole or an odd antihole. Considering first the former, let a and w denote the respective sizes of a largest stable set and a largest clique of G ;also let n = I V 1, where V is the vertex set of G. Certain direct consequences of minimal imperfection are immediate, e.g., that G is connected and that a 2 and w 3 2. From Lovasz’ characterization of perfect graphs [7] it follows easily that n=aw+1

(1)

for each u E V, V - u (denoting the set V - { u } ) may be partitioned into cliques of size w and into stable sets of size a.

(2)

and

A clique of size w (stable set of size a )is termed an w-clique (a-stable set). Padberg [9] has established:

G contains exactly n w-cliques and exactly n a-stable sets.

(3)

To each w-clique C in G there corresponds a unique a -stable set S in G such that C n S = 0.

(4)

The n x n incidence matrix of w-cliques with vertices of G is nonsingular with all row and column sums equal to w.

(5)

Straightforward consequences of (3)-(5) are the further properties (see [3,18]): The partitions described in (2) are unique.

(6)

For each u E V, the w-cliques containing u correspond (in the sense of (4))to the a-stable sets which partion V - u.

(7)

Additional properties of this type are described in [3, 11, 181. Next consider aspects of odd holes and odd antiholes which characterize them among minimal imperfect graphs. For example, recalling that G is minimal imperfect, clearly w = 2 implies

G is an odd hole and (8)

a = 2 implies G is an odd antihole.

Following [4] we denote by C : the graph with vertices uo, u l , . . . ,u,-, and edges {u,, a+,}, where 0 5 i 5 n - 1, 1 S j 5 k, and the index i + j is to be computed here (and below) modulo n. For n = 5,7,. . . , odd holes are of the form Ci with k = 1 and odd antiholes are given by k = (n - 3)/2. The following result (see [lo, 14, 171) shows that when the cliques of a minimal imperfect graph are ‘circular’ as in C:,then the graph must be an odd hole or an odd antihole. If G is isomorphic to C:for some k or an odd antihole.

3

1, then Gsis an odd hole

(9

The Strong Perfect Graph Theorem

163

This result has been strengthened by Chvdtal [4]:

If G contains a (spanning) subgraph isomorphic to Cf:for some k 2 1, then G is an odd hole or an odd antihole.

(10)

It is evident that the symmetry attributed minimal imperfect graphs by ( 5 ) is suggestive of the symmetry of graphs of the form Ci. Property (3,however, as well as properties (3), (4), (6) and (7), also holds (see [3, 181) for any graph which is parfitionable in the sense of (1) and (2). Furthermore, imperfect partitionable graphs which are not minimal imperfect have been given in [3,51. Thus a proof of Conjecture 1 based on (lo), i.e., showing that a minimal imperfect graph must contain a spanning subgraph isomorphic to Ci, should require the use of deeper information about minimal imperfection than that embodied in (5). In this paper we present the following strengthening of (10) which is directly linked to the partitioning information of (2). Theorem 1. If, for each v E V, the partition of V - v into a-stable sets has (at least) two members containing a single neighbor of v, then G is an odd hole or an odd antihole. Note that if for any v E V, the partition of V - v into a -stable sets has two members, S1 and S2, each containing a single neighbor of v, say v1 and u2 respectively, and v1 and v 2 are not adjacent, then G must be an odd hole. This follows from the fact that (6) implies that the subgraph H generated by S , U Sz U {v} is connected; hence a shortest cycle in H containing u, v1 and u2 is an odd hole. For certain classes of graphs the hypothesis of Theorem 1 is easily verified. A graph is K1,,-free if it does not contain

as an induced subgraph. Parthasarathy and Ravindra [12] have validated Conjecture 1 for K,,-free graphs. If the minimal imperfect graph G is KI.3-free, then the neighbor set of each vertex may be partitioned into two cliques, implying that each vertex has at most 2 0 - 2 neighbors. Since for each u E V, V - v partitions into w a-stable sets by (2), at least two of these a-stable sets contain only one of the 2w - 2 neighbors, and so validity of the hypothesis of Theorem 1 follows immediately. This method of obtaining the result of Parthasarathy and Ravindra was developed independently in [6, 191. Corollary 1 [12]. The Strong Perfect Graph Conjecture is valid for K1,3-freegraphs.

164

R. Giles et al.

Proof of Theorem 1. By (lo), it suffices to demonstrate that the vertices of G can so that for each 1 = 0,1,. . . ,n - 1, the vertex set be ordered uo, u,, . . . , {ul+,:0 =sj s w - 1) defines an w-clique of G; i.e., we will show that G contains the spanning subgraph C:i:l. Arbitrarily select uo E V and let {S,: 1 S j S w } denote the a-stable set partition of V - uo described in (2) (see Fig. 1). By hypothesis we may assume that uo is adjacent to only one vertex of SI, say u , E S , . Thus u 1 is adjacent only to uo among the vertices of the a-stable set Sl+m = SI- uI + uo (i.e., where we use the notation S + u for the set S U { u } ) and we have that {Sl+,: 1 S j 6 w } is the a-stable set partition of V - u1 (see Fig. 2). Continuing now inductively, suppose that k is fixed, 1 S k < n - 1, and that (i)-(iii) hold for 1 C 1 k. (i) uo, u , , . . . , uI are distinct vertices of G. is adjacent to u l . (ii) In S l + , = Sl - uI + ul-l only (iii) { S , + , :1 s j S w } is the a-stable set partition of V - uI. To complete the induction we verify (i)-(iii) for I = k + 1. Combining (ii) and (iii) for 1 = k with the hypothesis of Theorem 1 shows that some a-stable set St+,, 1 S j S w - 1, contains a single neighbor & + I of V k . Consider the case k < o depicted in Fig. 3 and first suppose u k + l E Sl+w. Since is in the a-stable set

00. . .

s,

...o

Fig. 1. a-stable set partition for V - u,.

Fig. 2. a-stable set partition for V - u , .

165


. . . s,,

...

..

s,,

. ..

...

s,,

..

...

s,

,

(00

6..

*

s,

*

..

*..

0) 0)

Fig. 3. a-stable set partition for V - uk,with k < w.

partitions for both V - uo and V - u k , it follows from (7) that the o-clique corresponding to & + I as in (4) contains both uo and u k . Thus uo and v k are adjacent and uo E Sl+w implies u k + l = uo. Substituting u k for uo in Sl+, yields the a-stable set partition of V - uo given by . . , S,, S1+, - v o + U k , SZ+-,. . .,s k + o } which is distinct from that specified by (iii) for I = 0. As this is in contradiction to (6), we conclude u k + l k ? S S l + w . A similar argument shows that u k + l $Z Si+-, for 1 S i S k - 1. Thus u k + l is a member of one of the a-stable sets &+I,. . . ,S,, and no generality is lost by assuming u k + 1 E S k + l . Note also that in this case (i) clearly holds for I = k 1. For the case k 2 w, we obtain, arguing as above, that if u k + l E & + j , 2 s j s w - 1, then u k + l = u k - o - l + j , again leading to a contradiction of (6) for V - u k - - - l + j . Thus in this case also u k + l E s k + l and it is again clear that (i) holds for 1 = k + 1 (see Fig. 4). Defining Sk+I+, = - u k + I + O k , we see immediately that (ii) holds for 1 = k + 1; (iii) follows by exchanging u k for u k + l in s k + 1 . An application of (iii) now shows that for 0s I s n - 1, the a-stable set S,+, is a member of the a-stable set partitions for

+

.

Fig. 4. a-stable set partition for V - u,, with k

3 o.


167

V - u l , .. . , V Thus it follows from (7) that for any choice of I, 0 s 1 s n - 1, the vertices uI,. . . ,ul+,-l constitute an w-clique of G, which completes the proof.

References [l] C. Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther Univ. Halle-Wittenberg Math.-Natur. Reihe 114 (1961). [2] C. Berge, Graphs and Hypergraphs (North-Holland, Amsterdam and American Elsevier, New York, 1973). [3] R.G. Bland, H.-C. Huang and L.E. Trotter, Jr., Graphical properties related to minimal imperfection, Discrete Math. 27 (1979) 11-22 (this volume, pp. 181-192). [4] V. Chvatal, On the strong perfect graph conjecture, J. Comb. Theory, Ser. B 20 (1976) 139-141. [5] V. ChvBtal, R.L. Graham, A.F. Perold and S. Whitesides, Combinatorial designs related to the perfect graph conjecture, Discrete Math. 26 (1979) 83-92 (this volume, pp. 197-206). [6] F.R. Giles and L. E. Trotter, Jr., On stable set polyhedra for K,,3-free graphs, J. Comb. Theory, Ser. B 31 (1981) 313-326. [7] L. Lovasz, A characterization of perfect graphs, J. Comb. Theory, Ser. B 13 (1972) 95-98. (81 H. Meyniel, On the perfect graph conjecture, Discrete Math. 16 (1976) 339-342. [9] M.W. Padberg, Perfect zero-one matrices, Math. Program. 6 (1974) 180-196. [lo] M.W. Padberg, Characterizations of totally unimodular, balanced and perfect matrices, in: B. Roy, ed., Combinatorial Programming: Methods and Applications (Reidel, Boston, MA, 1975) 275-284. [ l l ] M.W. Padberg, Almost integral polyhedra related to certain combinatorial optimization problems, Linear Alg. Appl. 15 (1976) 69-88. [I21 K.R. Parthasarathy and G. Ravindra, The strong perfect-graph conjecture is true for K,,,-free graphs, J. Comb. Theory, Ser. B 21 (1976) 212-223. [13] K.R. Parthasarathy and G. Ravindra, The validity of the strong perfect graph conjecture for ( K , - e)-free graphs, J. Comb. Theory, Ser. B 26 (1979) 98-100. [14] L.E. Trotter, Jr., A class of facet producing graphs for vertex packing polyhedra, Discrete Math. 12 (1975) 373-388. [15] L.E. Trotter, Jr., Line perfect graphs, Math. Program. 12 (1977) 255-259. [16] A.C. Tucker, The strong perfect graph conjecture for planar graphs, Canad. J. Math. 25 (1973) 103-114. [17] A.C. Tucker, Coloring a family of circular arcs, SIAM J. Appl. Math. 29 (1975) 493-500. [18] A.C. Tucker, Critical perfect graphs and perfect 3-chromatic graphs, J. Comb. Theory, Ser. B 23 (1977) 143-149. [19] A.C. Tucker, Berge’s strong perfect graph conjecture, Ann. New York Acad. Sci. 319 (1979)

53s.535.



A CHARACTERIZATION OF PERFECT MATRICES Manfred W. PADBERG New York University, New York, N Y 1ooo6, USA A zero-one matrix is called perfect if the polytope of the associated set packing problem has integral vertices only. By this definition, all totally unimodular zero-one matrices are perfect. In this paper we give a characterization of perfect zero-one matrices in terms of forbidden submatrices. The notion of a perfect zero-one matrix is closely related to that of a perfect graph as well as that of a balanced matrix as introduced by Berge. Furthermore, the results obtained here bear on an unsolved problem in graph theory, the Perfect Graph Conjecture.

1. Introduction

In this paper, we consider the polytope defined by the constraints of the following set packing problem: maxc x Axse xi = O o r 1 V j i E N = { l ,

..., n } ,

where A is an m X n matrix of zeroes and ones having no zero columns, eT = (1,. . . ,1) is the vector having all m components equal to one, and c is an arbitrary vector of reals. This problem has recently obtained much attention, see e.g. [l], [2], [6], [14], [18]. By (LP) we denote the linear programming problem obtained from (P) by dropping the integrality requirement on x. If the matrix A involved in problem (P) is totally unimodular [lo], then all basic feasible solutions to (LP) are integral, i.e., for any vector c the integer programming problem (P) can be solved as an ordinary linear programming problem. Generally, the matrix A encountered in (P) is not totally unimodular. Nevertheless, for certain matrices A the property that all basic feasible solutions to (LP) are integral, remains true (see Section 2 for relevant examples, also [5]). We call such matrices perfect zero-one matrices. Using some results from graph theory, we give a complete characterization of perfect matrices A in terms of forbidden submatrices. We give examples that show that - as in the case of balanced matrices, see [5] - it is not possible to characterize perfect zero-one matrices by means of forbidden determinantal values (as is the case for totally unimodular matrices). Indeed, given any natural number k, there exists a perfect zero-one matrix A such that it has a minor with determinant k. 169

M. W.Padberg

170

This paper is an abbreviated version of the original paper [15]; see also [16] where the main result of this paper is derived using a different proof technique.

2. Perfect Zero-One matrices

Let A be any m x n matrix of zeroes and ones having no zero column, and define the polytopes P and PI as follows:

P = { X E W I A X S ~ , X , ~ O ,j = 1,..., nl, P, = conv{x EW"1 AX c e, x, = o or 1, j = I , . . , n),

(2.1)

where e T= (1,. . . , 1 ) has m components all equal to one. The matrix A is called perfect if P = P I , i.e., if the polytope P defined in (2.1) has only integral vertices. Note that dim P = dim PI = n holds. Denote G the (intersection) graph associated with the matrix A, i.e., the nodes of G correspond to the columns of A and two nodes of G are linked b y an edge if the associated columns a' and a' of A have at least one + 1 entry in common, see e.g. [14]. A clique in G is a maximal complete subgraph of G. Let C denote the node set of any clique in G. Then by Theorem 2.1 of [14], the inequality

2 afx,

GI,

a; =

,=I

r

1 ifjEC (2.2)

0 if not

yields a facet of P I , i.e., a face of dimension n - 1 of P,. Clearly, every facet of the polytope PI is essential in defining P I .Hence it is a necessary condition for A to be perfect that A contain the incidence (row-) vectors of all cliques of the associated graph G. In order to characterize perfect matrices we can thus restrict ourselves to considering 'clique'-matrices, i.e., matrices A which contain the incidence vectors of all cliques of the associated graph G. Let A be any clique-matrix of size rn x n and let G be the associated graph. Let G denote the complement of G and denote B a clique matrix of G. Similarly to (2.1) define the polytopes Q and Q,, respectively, as follows: Q = { x E R " ( B x < ~ , x , z - o , j = 1 , ..., n),

I

0, = conv1x E R" BX s e, x, = o or I, j

= I,. .

.,n ) ,

(2.3)

where i?'=(l,. . . , l ) has all components equal to one and is dimensioned compatibly with B. Note that B has no zero column and that dim Q = dim QI = n holds. The vertices of Q, correspond to complere subgraphs of G, and vice versa. Furthermore, every maximal independent node set in G defines a clique of d (and vice versa). Consequently, there exists a (incomplete) 'duality' relation

171

A characterizationof perfect matrices

between the vertices of PI and the facets of Q (and, hence, between the vertices of Q l and the facets of P), see e.g. [ll].In the terminology of Fulkerson [7], Q (P, resp.) is the anti-blocker of Pl ( Q I , resp.). Let A be any clique-matrix of size m X n. By Theorem 1 of [ 6 ] ,A is perfect if and only if the associated graph G is perfect. Consequently, A is imperfect if and only if the graph G contains an induced subgraph of G’ which is almost perfect, i.e., G’ is imperfect, but every proper induced subgraph G’ is perfect, see [13], [15], [17]. Since induced subgraphs of G having k nodes correspond uniquely to m X k submatrices of A (and vice versa) we can make, without loss of generality, the assumption that G is an almost perfect graph. Denote a ( G ) the maximum cardinality of a stable (independent) node set in G and define C Y ( ~likewise ) for G. Lemma 1. Let A be a clique-matrix of an almost perfect graph G, LY = a ( G )and o = (Y ( G ) .Then xi < o provides a facet of QI and x = ( l / w ) e is a fractional vertex of P.

cY=,

+

Proof. Denote by e’ the row vector with n - 1 components equal to 1 and having a zero in the j-th component. Since G is almost perfect it follows from Theorem 1 of [13] that

e’x = a! for j = 1 , . . . , n. max XEP Define Hito be the halfspace given by

H, = { X E W ”I e ’ x s a } f o r j = 1 , ..., n. Hi for j = 1 , . . . , n or, equivalently,

Consequently P

n H’.

PCH=

j=l

By definition of e’, j apex at

x-

f f =-

n-1

=1,.

. . ,n, H is a pointed polyhedral convex cone with its

e.

By Theorem 1 of [13], we have that a ( G ) oa! (G) = n - 1 holds, since G is almost perfect. Hence it follows that Ai a + 1, so there exists u E V \ S. By (1.2b) there is a partition of V \ u into a w-cliques of G, which implies that two distinct vertices of the stable sets S are in a common clique, a contradiction. Thus I S 1 s a for all stable sets S in G. Similarly 1 Q 1 s w for all cliques Q in G.

Claim 2. For every clique Q in G there exists an a-stable set S in G such that QnS=@. Roof. For any o E Q there is a partition of V \ u into stable sets SL, S 2 , .. . ,S,. By Claim 1 10 \ u I < w - 1. Since each vertex in Q is in at most one of S , , S I , .. . ,S,, we have Q f l Si = 0 for some 1 S i 6 w.

In the case where the partitionable graph G is minimal imperfect, the following Claims 3-5 culminating in Proposition 6 are either due to Padberg [14] or follow easily from his results. In what follows 0, denotes the k-vector (0,. . . , O ) , l k denotes the k-vector (1,. . . , l), and the superscript t is the transpose operator. Claim 3. There are n cliques in G whose n and has all row and column sums equal to

X

n incidence matrix is nonsingular

w.

Graphical properties related to minimal imperfection

185

Proof. Let Q = { v l , . . . ,uw}be any o-clique of G. Let c, be the incidence vector of Q and for i = 1,.. . ,w, let Mi be the a X n clique-vertex incidence matrix of some partition of V \ ui into w-cliques. We then define C to be the matrix

Since each M , is an a X n matrix and n = a w + 1, C is an n X n matrix. Also note that the row and column sums of C are all w. Therefore y = (l/w)l,, is a solution of the system y C = 1,. We will prove that C is nonsingular by showing that y is the unique solution. Suppose that for some y # y we have yC = 1,. Since j j > 0, we lose no generality in assuming that y 2 0. For i = 1,. . . ,n let e, denote the ith unit vector, and let c, denote the ith row of C. By Claim 2 there corresponds to each c, an incidence vector a, of an a-stable set such that a, . c, = 0. Hence for any i = 1,. . . ,n we have a = (1, - c l ) .a, = [(y - e , ) C ] .a, = (y - er)(Ca:) =

c

,=1

y,(c, * a z ) + ( y , - l)(c, . at).

f#I

But a = ( n - l)/w, c, a, = 0 or 1 for i# j , and c, - a, = 0. Therefore, since y we have

2 y j 3 -n. - 1 i=l

w

I#i

On the other hand, 11 =

1, * 1, = (yC). 1" = y(C1',) = y . ( w l . )

Hence we have that

From (2.1) and (2.2) we see that y

= (l/w)ln.

=w(y

*

1.1.

20

R.G. Bland et al.

186

It is important to note that the incidence vectors ai in the proof of Claim 3 must in fact satisfy ai cj = 1 for all i f j . We let Sidenote the a -stable set with incidence vector ai and denote by A the n X n matrix whose ith row is ai, i = l , ..., n. Claim 4. Let E and I denote the n X n matrix of ones and the n X n identity matrix, respectively. Then CA' = E - I, so C-' = ( l / w ) E - A', i.e., C-' has as the entry in its jth row and ith column w-1

, if uj E Si, otherwise.

-

Proof. This follows immediately from the fact that ai ci CE = WE,as implied by Claim 3.

= 1 for

i# j and from

Claim 5. G has exactly n w-cliques.

Proof. Let c be the incidence vector of any maximum clique in G. Then y = cC-' is the unique solution of the system yC = c. From Claim 4 we see that Yi

0 if ai * c = 1, 1 if ai - c =O.

=

Also w

c

i=l

yi

=y

(a:=)(yC). 1" = c . 1,

cT=,

= w,

implying that y, = 1. Thus y = ( y , , . . . ,yn) has y, for all j# i, so c = c,.

=

1 for some i and y, = 0

It follows from Claim 5 and the proof of Claim 3 that the a-stable set S of Claim 2 having S n Q = 0 for some w-clique Q is unique. The results above are summarized in the following proposition.

Proposition 6. Let a 3 2 be integers and let G be a graph that satisfies (1.2). Then G has exactly n = (YW+ 1 (maximum) w-cliques. There is a 1-1 correspondence between w-cliques and a-stable sets, pairing each o-clique Q with the unique a -stable set S having S n Q = 0. The incidence martix of @-cliques with vertices is nonsingular with all row and column sums equal to w. Furthermore G, the complement of G, and 6 = w. W = (Y also satisfy (1.2).


187

Since minimal imperfect graphs are partitionable, Proposition 6 includes Padberg's property (1.3) for minimal imperfect graphs. Further properties follow from those outlined above. These will be stated in terms of cliques; analogous properties obviously hold for stable sets. Claim 7. The partition of the vertices of G\vi into w-cliques is unique. Proof. A partition of the n - 1 = a w vertices of G \ v, into w -cliques represents a solution of the system yC = (1, - e!). Since C is nonsingular, y is unique. Claim 8. For v E V, the a a-stable sets that contain v correspond (in the sense of Proposition 6) to the a w-cliques in the unique partition of the vertices of G \ u into w-cliques. Proof. Let v E V and let {Ql, Q 2 , . .. , Q,} be the unique partition of V \ u into w-cliques. Recall that for 1 i G a, Si is the unique a-stable set having S, fl Q, = 0. Since IS, n 0,I = 1 for all j # i, we have 1 S, n (V\{v})i = a - 1. But 1 S, I = a, so it must be that v E Si. Let B be the p x n incidence matrix of all maximal cliques of G ;p 2 n since G may have maximal cliques Q with 1 Q I < w. Let R : denote the nonnegative orthant of R " and let BG and PG be the polytopes

PG = { x E R : : Bx'G l;} and

BG

= convex

hull

{X

E {0,1}" : B x ' S 1;).

Clearly BG C PG and the set of integer extreme points of PG is the same as the set of extreme points of BG;it is precisely the set of incidence vectors of stable sets of G. Furthermore, since C is a nonsingular submatrix of B and Cx' = 1; for x = (l/w)l,,, it follows that (l/w)ln is an extreme point of PG. Padberg proved the next claim. Claim 9 [15]. If G is minimal imperfect, then the point (l/w)ln is the unique fractional extreme point of PG. The following lemma, which can be easily proved, will be useful in the proof of Claim 9. Lemma 9.1. Let 61,.. . ,S. l/S1+. * + 1/S, 3 n / s .

-

be

positive

with

+

S , +. . . 6, s nE.

Then

R. G. Bland et al.

188

Proof of Claim 9. Note that every x E Pc must satisfy Cx' s lt,, or equivalently, s' = 1: - Cx' 3 From Claim 4 C ' = ( ( l / w ) E - A '), and as already observed, C-'l: = (l/w)lk. Let LT = 1, . s. Then 1

c)w

lL+ A'S'.

(2.3)

Observe that I1

I;x

= (l -V -+C Ta , w

and so 1,

' X

z a

if and only if

CT 4

1.

Since s 3 0" wc have u 2 0, and from (2.3) and (2.4) we see that 1, . x 3 a if and only if I is a convex combination of the extreme point ( l / w ) l n of Pc and the extreme points of PG corresponding to a-stable sets in G. Suppose that x = (x,, . ..,x,) is a fractional extreme point of PG. Then clearly 1 > x, > 0 for j = 1.. . . , n, since G is minimal imperfect. Hence there exist n cliques of G yhose n X n incidence matrix D is nonsingular and has Dx' = 1:. Let 1,D = 6 = (S,,. . . , a n ) and for j = 1,. . . ,n let 8' = (a;, . . . ,a',,) have 6: = 0 and 6: = 6, for all k # j . Since G is minimal imperfect, the linear form 6 ' . y achieves a maximum over PG at an extreme point y = y' that is the incidence vector of a stable set in G. Since x # y' E PG and D is nonsingular, we have that n = 6 . x > 6 . v'. and since 6 . y' is an integer, n - 1 2 6 . y' 3 6' . y'. Furthermore, because y = y' maximizes 6' . y over y € PG, we see that n

-

1 3 6' . x = 6 . x - 6'X' = n - S'X,.

Thus 6,x, 3 I , and since 6 . x = n we obtain 6,x, = 1. or x, = l / & for j = 1,. . . ,n. Also note that 6, + . . . + 6, d nw, since w ( G ) = w and each row of D is the incidence vector of a clique in G. I t now follows from Lemma 9.1 that

I,, x

=

1/al

+ . . . + 1/6,

2 n/w

> a.

Thus (2.4) implies that x = (l/w)l,. I t is easy to derive from Claim 4 another result of Padberg [15]: the fractional extreme point ( l / w ) l n is adjacent to each of a l , . . . ,a, in Pc. While the proof of Claim 9 uses minimal imperfection (the result obviously fails if G is not minimal imperfect), the following related result is true for all partitionable graphs G.

Claim 10. The subgraph of G induced by the symmetric difference of any two a -stable sets is connected.

Graphical properfies related to minimal imperfection

189

Proof. Suppose that S, and S,, i f j, are two maximum stable sets; let a, and a, be the associated incidence vectors. Let S,,u S,, = S, \ S, with S,l n S,, = 0 and let

S,, u S,, = S, \S, with S,, n SJ2= 0, where S,, U S,, is a component of the subgraph induced by ( S , \ S , ) U (S, \ S,). Thus S,lU S,, and S,, U S,, are stable sets, and consequently S=S,,US,,U(S,nS,)

and

S=S,,uS,,U(S,nS,)

s,

are a-stable sets. Let 6 and d be the incidence vectors of and respectively. Now, 6 + Ci = u, + a,, so these four vectors are linearly dependent. Hence, by the linear independence of the y1 incidence vectors of a-stable sets, it must be that either a, = 6 and a, = Ci or a, = 6 and a, = 6. Therefore the subgraph induced by (S, \S,) U (S, \ S , ) is connected. Chvatal [4]has shown that if B is a polytope that is the convex hull of the incidence vectors of the stable sets in a graph G, then two extreme points of B are adjacent if and only if the symmetric difference of the corresponding stable sets induces a connected subgraph of G. Thus, Claim 10 together with Chvatal's result implies that each pair of extreme points of BG corresponding to a -stable sets is adjacent in BG.Claim 9 can then be used to prove that the extreme points corresponding to the n incidence vectors of a-stable sets in any minimal imperfect graph G and the unique fractional extreme point ( l / w ) l n arc mutually adjacent extreme points of PG;i.e., they form an n-simplex in the skeleton of PG.

3. Examples

In light of the stringent symmetries embodied in (1.3), one might hope to prove Conjecture 2 by demonstrating that any imperfect graph with property (1.3) has a spanning subgraph isomorphic to C:l,' where a ( G )= a, w ( G ) = w . We will now describe two counterexamples, partitionable graphs that have no spanning subgraphs of the form CZ;,. Consider the graph G I of Fig. I , having n ( G , )= 10 and a ( G , )= w ( G , )= 3 . Table 1 describes for each u E V(GI) partitions of V ( G I ) \u into 3-cliques and 3-stable sets. Thus G I is a partitionable graph and, therefore, satisfies (1.3). Yet GI has no spanning subgraph isomorphic to C;", as is evident from the fact that there is only one 3-clique Q' in G I ,that has I Q n Q' 1 = 2 , where Q = {1,2, A } ,a 3-clique of GI. A 5-hole is induced by the subset { I , 3,5,7,9} of V(G,).

R.G. Blander al.

1

6 Fig. I . The partitionable graph G , .

Table 1. Partitions of G , \ u Vertex deleted

1

7

3 4

S 6 7 8 9 A

Clique partition 234, 678, 59A 345. 89A, 167 12A, 456, 789 123. 678, 59A 234, 89A. 167 12A,345, 789 123. 456, 89A 234, 167, 59A 12A,345. 678 123. 456. 789

Stable set partition 369, 41AA,258 369, 47A, 158 47A, 158, 269 158, 269, 37A 269, 37A. 148 37A, 149. 258 149, 36A. 258 149, 36A, 257 36A, 257, 148 369, 257, 148

Consider the graph G2 of Fig. 2, which has n(G2)= 13, w ( G 2 )= 3, and (Y (G2)= 4. Table 2 lists partitions for each u E V(G2)of V(G2)\u into 3-cliques and into 4-stable sets. The 3-clique 0 = {1,5,9} in G2 has I 0 r l Q ' I # 2 for all 3-cliques Q' in G2, so the partitionable graph G2 has no spanning subgraph isomorphic to C%. G2 contains a 5-hole induced by {1,2,5,6,8} and a 7-hole induced by {2,3,4,7,9, B, D}. Note that the pair {3,8} is in neither a maximum clique nor a maximum stable set in G I .The pairs (6, A } and {S, C} have the same open status in G2.Since (1.2) depends only on the cliques and anticliques of maximum size, any such open pair can be added as an edge without disturbing property (1.2). Among the partitionable graphs of the form C:,only odd holes and odd antiholes have no such open pairs.


191

1

Fig. 2. The partitionable graph G,.

Table 2. Partitions of G,\ u Vertex deleted 1

2 3 4

5 6 7 8 9 A B C D

Clique partition

Stable set partition

23A, 456, 789, BCD lCD,345, 678, 9AB 12D, 456, 789, ABC 159, 23A, 678, BCD lCD,23X, 467, 9AB 12D, 345, 789, ABC lCD,238. 456, 9AB 159, 23A,467, BCD 12D, 345, 678, ABC 159, 238, 467, BCD ICD, 23A, 456, 789 12D, 345, 678, 9AB 159, 238, 467, ABC

257B, 369C, 48AD 148B. 369C, 57AD 148B, 26YC, S7AD 137B. 26YC, SXAD 137B, 269C, 48AD 137B, 249C, 58AD 136B, 249C. S8AD 136B, 249C, 57AD 136B. 2S7C, 48AD 148B, 257C. 369D 148A, 257C, 369D 148A, 257B, 36YD 148A, 257B, 369C

Much of our work here, including the examples G I and Gz, was first announced in [lo]. The authors have recently learned of the related work in [6], where explicit procedures for constructing partitionable graphs are given. G I and Gz are among the partitionable graphs discussed in [6].

R.G. Bland et al.

192

References [ I ] C. Bcrgr. Firbung von Graphen, deren samtliche hzw. deren ungerade Krcise \tarr sind. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe I 14 (1961). 121 C. Berge. Graphs and Hypergraphs (North-Holland, Amsterdam. 1973). 131 C. Berge. The history of the perfect graph conjecture (to appear). 141 V. Chvital. O n certain polytopes associated with graphs. J . Comb. Theory, Scr. B 18 (1975) 138- 154. 151 V. Chvatal. O n the strong perfect graph conjecture. J. Comb. Theory, Ser B 20 (1Y76) 139-141. 161 V. Chviral, R.L. Graham. A.F. Perold and S. Whitcsidrs, Combinatorial designs relatcd t o the strong perfect graph conjccture. Centre d e Recherches Mathematiques. Univcrsiti. de Montreal. Publication 278. 171 D.R. Fulkerson. Blocking and anti-blocking pairs o f polyhedra. Math. Program. 1 (1971) 168- 194. [XI D.R. Fulkerson. Anti-blocking polyhedra. J. Comb. Theory I2 (1972) S(k71. 191 D.R. Fulkcrson. On the perfect graph theorem. in: T.C. H u and S.M. Robinson. cds.. Mathematical Programming (Academic Press, New York, 1973) 69-76. 101 H.-C. Huang. investigations on combinatorial optimization, Ph.D. Thesis. School of Organization and Management, Yale University (School of O R / I E , Cornell University, Technical Report No. 308 (1976)). [ 111 1.. Lovrisz. Normal hypergraphs and the perfect graph conjecture. Discrete Math. 2 (1972) 253-267 (this volume, pp. 29-42). [ 121 L. Lovasz. A characterization of perfect graphs, J . Comb. Theory. Scr. B 13 (1972) 95-98. [ 131 H. Meyniel. O n thc perfect graph conjecture. Discrete Math. I6 (1976) 339-342. [ 141 M. Padberg. Pcrfcct zero-one matrices, Math. Program. 6 (1974) IX(k196. 1 IS] M. Padberg. Almost integral polyhedra rclated to certain combinatorial optimization problcms. Linear Algebra & Appl. IS (1976) 69-88. [ 161 K.R. Parlhasarathay and G. Ravindra, The strong perfect graph conjecture is true for K,,,-free graphs. J. Comb. Theory, Ser. B 21 (1976) 212-223. [ 171 K.R. Parthasarathay and G. Ravindra, The validity of the strong perfect-graph conjccture for (K4- *)-free graphs, J. Comb. Theory, Ser. B 26 (1079) 98-100. [ 181 H. Sachs. O n the Berge conjecture concerningperfect graphs. in: Combinatorial Structures and Thcir Applications (Proceedings of the Calgary International Conference, 1969) (Gordon and Breach, New York, 1970) 377-384. 1191 L.E. Trotter, Jr.. Line perfect graphs. Math. Program. I 2 (1977) 255-259. [ZO] L.E. Trotter. Jr.. A class of facet producing graphs for vertex packing polyhedra. Discrete Math. 12 (1975) 373-388. [21] A.C. Tucker. The strong perfect graph conjecture for planar graphs, Canad. J. Math. 25 (1973) 103-114.

(221 A.C. Tucker, Coloring a family of circular arcs, S l A M J. Appl. Math. 29 (1975) 493-500. [23] A.C. Tucker. Critical perfect graphs and perfect 3-chromatic graphs. J. Comb. Theory, Ser. B 23 (1077).

Annals of Discrete Mathematics 21 (1984) 193-195 0 Elwvicr Science Publishers B.V.

AN EQUIVALENT VERSION OF THE STRONG PERFECT GRAPH CONJECTURE

v. CHVATAL Sciiool

01

Computer Scieiice. McGill Uiiiuenity, Monrrenl.

C~iiiiiih

When G is a graph, a ( G )denotes the largest size of a stable (independent) set of vertices in G and w ( G ) denotes the largest size of a clique in G. A graph is called perfect if each of its induced subgraphs H is w(H)-colorable. This notion comes from Claude Berge [I], [ 2 ] , who conjectured that every minimal imperfect graph is isomorphic to a cycle whose length is odd and at least five or to the complement of such a cycle. This conjecture is known as the Strong Perfect Graph Conjecture. An equivalent version of the Strong Perfect Graph Conjecture was proposed in [ I ] ; the purpose of this note is to present the argument from [ 11 in a slightly different form. We shall denote by C: the graph with vertices u I , u2,. . . , u,, such that u, is adjacent to u, if and only if 1 i - j I d k, with subscript arithmetic modulo n. Theorem. Let G be a graph which contains a spanning subgraph isomorphic to Czi:l with a = a ( G ) and w = w ( G ) . If a 3 3 and w a 3 then G is not minirnal imperfect.

Proof. Writing n = a w + 1, we may enumerate the vertices of G = (V, E ) as uI, u?, . . . , u,, in such a way that (with subscript arithmetic modulo n ) u,u,+, E E

for all i and for all j

=

1,2,. . . , w - 1.

=

1,2,. . . , a

(1)

Now we may assume that

U , ~ , + ,E< ~ Efor all i and for all j

-

1;

otherwise the subgraph induced by u,, v , + ~. ., . , u,+,, is not w-colorable, and so G is not minimal imperfect. Next, writing w, = u,, for all i = 1 , 2 , . . . , n, we record ( 2 ) as

w , w , + , E E for all i and for all j 193

= 1,2,..

., a

-

1.

(3)

V.Churital

194

From (3), we conclude easily that every clique of size w consists of vertices I)u for some i. To put if differently,

w , , w , ~ .~. ,, w,,,,, .

every clique of size w consists of consecutive u,'s.

(4)

Similarly, (1) implies that every stable set of size a consists of consecutive wi's. If P stands for the set of vertices u,, u2, uO+1, uw+3 and 2,3,. . . ,a - 1 then (4)implies easily that

(5) uku+2

every clique of size w includes a vertex from F.

with k

=

(6)

Similarly, if Q stands for the set of vertices W ~ + I - ~ ~ w, ~ -W ~~ + Z~ - ~. w~ ~, ~ + I and -~ wLu with k = 0, 1,. . . , w - 3 then ( 5 ) implies easily that every stable set of size a includes a vertex from Q. Note that uz = Since a 3 3 and w

(7)

uu+3 = W n + l - 3 o , uzw+2= w ~ + and ~ - u,~ =~wo. five distinct vertices belong to both P and Q. Hence

LL,+~ = w.+~-,,

3 3, these

J Pu 01s a + w

-

1.

(8)

The rest is easy: it sutiices to consider the subgraph H of G induced by all the vertices outside F and 0. By (6), (7) and (S), we have w(H)sw-l,

a ( H ) s ( ~ - l and

JHl>(a-l)(u-l),

respectively. It follows that H is imperfect, and so G is not minimal imperfect. 0

Corollary. The Strong Perfect Graph Conject~treis true if and only if euery minimal imperfect graph G contains a spanning subgraph isomorphic to Czi;, wirh a = a ( G ) and w = w ( G ) . Proof. To establish the 'if' part, consider a minimal imperfect graph G. By the assumption, G contains a spanning subgraph isomorphic to Czil, with LY = a ( G ) and w = w ( G ) ; by the above theorem, a s 2 or w 5 2 . It is an easy exercise to show that G is isomorphic to Czo+l in case w S 2 and that G is isomorphic to the complement of C2w+l in case a s 2 . To establish the 'only if' part, it suffices to observe that the cycle of length 2 k + 1 with k 2 2 is isomorphic to C:ull. with a = k, w = 2 and that the complement of this cycle is isomorphic to Czi;, with a = 2, w = k. 0

An equivalent version of the Srrong Perfect Graph Conjecture

195

References [l] C. Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. 2. Martin-Luther Univ. Halle-Wittenberg 114 (1961). [2] C. Berge, Sur une conjecture relative au probltme des codes optimaux, Cornrn. 136me assemblee genCrale de I’URSI, Tokyo (1962). [3] V. Chvatal, On the Strong Perfect Graph Conjecture, J. Comb. Theory 20 (1976) 139-141.



COMBINATORIAL DESIGNS RELATED TO THE PERFECT GRAPH CONJECTURE*

v. CHVATAL School of Computer Science, McGill University, Montreal, Canada

R.L. GRAHAM Bell Laboratories, Murray Hill, New Jersey, U.S.A.

A.F. PEROLD Harvard Business School, Boston, Massachusetts, USA

S.H. WHITESIDES School of Computer Science, McGill Uniuersity, Montreal, Canada

Introduction

Our graphs are ‘Michigan’ except that they have vertices and edges rather than points and lines. If G is a graph then n = n ( G )denotes the number of its vertices, a = a ( G ) denotes the size of its largest stable (independent) set of vertices and w = w ( G ) denotes the size of its largest clique. The graphs that we are interested in have the following three properties: (i) n = a w + l , (ii) every vertex is in precisely a stable sets of size a and in precisely w cliques of size w , (iii) the n stable sets of size a may be enumerated as S , , Sz,. . . , S, and the n cliques of size w may be enumerated as C1,Cz,. . . , C, in such a way that S, n C, = 0 for all i but S, n C,#B whenever i # j . We shall call them (a,w)-graphs. This concept, contrived as it may seem at first, arises quite naturally in the investigations of imperfect graphs; we are about to explain how. In the early 1960’s, Claude Berge [I], [ 2 ] introduced the concept of a perfect graph : a graph is called perfect if and only if, for all of its induced subgraphs H, the chromatic number of H equals w ( H ) . Berge formulated two conjectures concerning these graphs: * Reprinted from Discrete Math. 26 (1979) 83-92. 197

(I) a graph is perfect if and only if its complement is perfect; (11) a graph is perfect if and only if it contains no induced subgraph

isomorphic either to a cycle whose length is odd and at least five or to the complement of such a cycle. The concept of a perfect graph turned out to be one of the most stimulating and fruitful concepts in modern graph theory. The weaker conjecture (I), proved in 1971 by Lovasz [lo], became known as the Perfect Graph Theorem. The stronger conjecture (II), still unsettled, is known as the Perfect Graph Conjecture. A graph is called minimal imperfect if it is not perfect itself but all of its proper induced subgraphs are perfect. Clearly, every cycle whose length is odd and at least five is minimal imperfect, and so is its complement. The Perfect Graph Conjecture asserts that there are no other minimal imperfect graphs. The first step towards a characterization of minimal imperfect graphs was made again by Lovasz [ I l l : every minimal imperfect graph satisfies n = a w + 1. It follows from this that, in a minimal imperfect graph G, for every vertex u E G, the vertex set of G - u can be partitioned into a cliques of size w, and into w stable sets of size a. Further refinements along this line are due to Padberg [12]: every minimal imperfect graph is an (a,w)-graph. (Bland et al. [3] strengthened Padberg's result by proving that every graph satisfying (1) in an (a,w)-graph.) Hence characterizing (a,w )-graphs might help in characterizing minimal imperfect graphs. It is easy to construct (a,w)-graphs for every choice of a and w such that a 2 2 and o 3 2: begin with vertices uI, u 2 , .. . , and join 0, and u, by an edge if and only if I i - j I s w - 1, with subscript arithmetic modulo aw + I . The resulting graph, denoted by CZil, is an (a,w)-graph. If w = 2 then CEiI is simply the odd cycle Czo+,; if (Y = 2 then C2i:l is the complement of the odd cycle C,,,,. If a 2 3 and w 3 3 then C Z : , contains several pairs of nonadjacent vertices u, w such that joining u to w by an edge destroys no stable set of size a and creates no new clique of size w. Hence the graph obtained by joining u to w is again an (a,w)-graph. However, calling this graph new smacks of cheating: the structure of the largest stable sets and of the largest cliques has remained unchanged. To avoid such quibbling, we shall consider normalized ( a ,w)-graphs in which every edge belongs to some clique of size w. (As we shall see in a moment, every (a,w)-graph contains a unique normalized (a,[.')-graph.) The purpose of this note is to present two different methods for constructing normalized (a, w)-graphs other than C ~ , The ~ ~smallest . of these graphs is the (3,3)-graph shown in Fig. 1. (This graph and the (4,3)-graph of Fig. 4 were

Combinatorialdesigns related to the Perfect Graph Con]ecture

199

Fig. 1.

independently presented in [3] as examples of (a,o)-graphs different from CZll; see also [9].) The problem of characterizing all the normalized (a,w)-graphs can be given at least two additional interpretations. First, with each (a,o)-graph we may associate two zero-one matrices X , Y of dimensions n x n such that the rows of X are the incidence vectors of the stable sets S , , Sz, . . . ,S, and the columns of Y are the incidence vectors of the cliques C1,C,, . . . ,C.. If I denotes the n X n identity matrix and if J denotes the n X n matrix filled with ones then clearly

JX = XJ =aJ, JY = YJ = wJ, X Y

=J

- I.

(2)

In the terminology of Bridges and Ryser [4],the matrices X and Y form an ‘(n,O,l)-system on a, o’.Conversely, with each pair of zero-one matrices X, Y satisfying (2), we may associate a graph G with vertices u l , uz, . . . , u, such that u, is adjacent to u, if and only if Yri= Ysi= 1 for some j . Let us show that G is a normalized (a,o)-graph. To begin with, each column of Y generates a clique of size w in G and, since X Y is a zero-one matrix, each row of X generates a stable set of size a in G. To show that G has no other cliques of size w, consider an arbitrary clique of size w and denote its incidence vector by d. Clearly, Xd is a zero-one vector. In fact, since J ( X d ) = (JX)d = aJd, the vector Xd has a w = n - 1 ones and one zero. Hence Xd is one of the columns of J - I = X Y . Finally, since X is nonsingular, d must be a column of Y.A similar argument shows that every stable set of size a in G arises from some row of X . Hence G is an (a,w)-graph; since each edge of G belongs to some clique of size w, G is also normalized.

V. Chvcital er al.

200

The matrix interpretation makes it easier t o clarify the role of normalized (a,w)-graphs. Consider an arbitrary (a,w)-graph G and delete all those edges which belong to no clique of size o.To show that the resulting graph H is an (a,w)-graph, it will suffice to show that every stable set of size a in H was also stable in G. Beginning with G, define X and Y as above; in addition, let d denote the incidence vector of an arbitrary stable set of size a in H. Since the cliques of size o are the same in G and H, the vector d Y is zero-one. Since (dY)J = d ( YJ) = wdJ, the vector dY is one of the rows of X Y . Since Y is nonsingular, d is one of the rows of X , which is the desired conclusion. Hence H is the unique normalized (a,w)-graph contained in G. I n the next section, we shall make use of the fact that the equations (2) imply

YX

= X - ' X Y X = X-'(J

- I ) X = X-IJX

- I = J - I.

(The above observations are due to Padberg [12].) Before proceeding, let us point out a simple fact which may be useful in constructing (a,w>graphs. For the moment, we shall refer to each pair of matrices ( X , Y ) satisfying (1) as a solution. Now, let r and s be positive integers such that r + s = n. Let A, A * be n X r matrices, let B, B * be n X s matrices, let C, C* be r x n matrices and let D , D * be s x n matrices. Finally, let us write

XI = ( A ,B * ) , Xz= ( A *, B ) , X 3 = (A,B ) , X4 = (A *, B * ) and

We claim the following: if ( X I ,Y l ) ,( X 2 ,Yz),(X,, Y 3 )are solutions then ( X + Y4) is a solution. The proof is straightforward: since

+ B * C* = J - I, A * C* + B D = J - I,

XI Y l = A C

Xz Yz

X,Y,=AC+BD=J-I, we have AC

= A * C * , BD = B*D*

X4Y4 = A * C * + B * D * = J

and so

- I.

Similarly, the equations JX4= X4J = (YJ and JY, = Y,J routinely. It may be also interesting to note that:

= wJ

follow quite

if ( Y , ,X I ) ,( Y2,X z ) ,( Y3,X,)are solutions then ( Y 4 ,X4)is a solution.

Combinatorial designs related to the Perfect Graph Conjecture

20 1

The point is that the equations

imply

X k y k

=J

- I for each k = 1,2,3. Now X4Y4= J

-I

as above, and so

Y4X4= J - I.

An alternative interpretation of ( a ,w )-graphs concerns a packing problem. With a slight abuse of the standard notation, let K , denote the directed graph on n vertices such that, for every ordered pair of vertices u and w, there is a (unique) directed edge from u to w . Similarly, let Ka,,denote the complete bipartite graph in which each edge is directed from the a-set. As above, let n stand for aw + 1. We claim that normalized ( a ,@)-graphs correspond to partitions of the edge-set of K,, into n disjoint copies of Ka.u.With every such partition, one may associate n X n matrices X , Y such that the j-th column of X is the incidence vector of the a-set of the j-th copy and such that the i-th row of Y is the incidence vector of the o-set of the i-th copy. It is not difficdt to verify that these matrices satisfy (2), Conversely, with every pair of zero-one matrices satisfying (2). one may associate a partition of Kn into n disjoint copies of Km,w by making the directed edge uiui belong to the k-th copy if and only if X t k = y k j = 1. Incidentally, if the directions of the edges are ignored then these partitions become covers of the undirected K, by n copies of undirected Kn.w such that each edge is covered precisely twice. Designs of this kind have been studied by C . Huang and Rosa [6], [7], [8]. Finally, let us return to the link between the problem of characterizing (a,&)-graphs and the Perfect Graph Conjecture: it is not clear that a solution to the former would indeed help to settle the latter. In fact, Tucker [13] succeeded in proving the Perfect Graph Conjecture for all graphs G with w ( G ) = 3 without characterizing ( a ,3)-graphs. By virture of Padberg’s theorem the Perfect Graph Conjecture may be stated as follows: every ( a ,w)-graph G with a 3 3 and w 2 3 contains a smaller induced imperfect graph. We shall say that an (a,w)-graph G is of type I if it contains a set W of a + w - 1 vertices such that W n S # P, for all stable sets of size a and W f l C# 0 for all cliques of size w . Otherwise we shall say that G is of type 11. It is easy to see that every ( a ,@)-graph of type I contains a smaller induced imperfect graph (namely, the graph G - W with ( a - l ) ( w -1)+1 vertices and a(G - W ) S a - 1, w ( G - W ) S w - 1). Hence the Strong Perfect Graph Conjecture would follow if every (a,w)-graph with a 3 3 and w 3 3 were of type I. Unfortunately, this is not the case: the (4,4)-graph constructed in Section 2 of this paper is of type 11. (In [ 5 ] , it has been shown that every CZ:, with a 2 3 and w a 3 is of type 1.1

V.Chvatal er al.

202

1. The 6rst method Each graph C&'lb+lcan be seen as arising from Ctill by a simple construction which, vaguely speaking, leaves most of the graph unchanged and increases the total number of vertices by w. We are about to show that the same construction applies in a more general setting: if some set of 2 0 - 2 vertices of an (a,w)-graph G induces a subgraph resembling a piece of Cz-'then a simple local change in G creates an (a + 1, w)-graph H. More specifically, the properties required of the 2w - 2 vertices uI, u 2 , .. . ,u2w-2in G are that each Of the Sets c k = { u k + l , u k + 2 , . . . , with k = 0,1,. . .,w - 2 is a clique, and that for each k = 2 , 3 , . . . ,o - 1, either c k - 1 is one of the a cliques partitioning G - u k - I or else Ck-2is one of the a cliques partitioning G - u o + k - l . The graph H has w new vertices ul,u 2 , .. . ,u, in addition to the old aw + 1 vertices of G. The adjacencies in H are best described in terms of its cliques of size W .First of all, we delete edges which belong to the w - 1 cliques C,specified above and no others. Each c k is replaced by two cliques,

c;=

{0k+lr uk+2,.

..

9

&-I,

a2,.

..

3

ak+I},

c ~ = { u k + 2 , u k + 3 , . . . . , a ~ , v o , . . . , v w + k } .

Finally, we introduce the clique C*= {al,u 2 , .. . ,u w } .In case w = 3, the passage from G to H is schematically illustrated in Fig. 2 . Before proving that H is indeed an (a +l,w)-graph, let us consider a few examples. To begin with, take G = C: and consider four consecutive vertices in the natural cyclic order. If these four vertices are labeled as uI, u2, u3, v4 then H = C&;however, if they are labeled as ol, v3, u2, u4 then H is the graph of Fig.

H a2 Fig. 2.

a3

v3

v4


203

1. Next, let G be the graph of Fig. 1. The three choices (Ul, uz,

0 3 , u4) =

@,I,2,3),

( V l , UP, u3,u4)

= (2,0,1,9),

(01, u 2 , u 3 , u 4 )

= (3,1,2,0),

lead to the (4,3>graphs shown in Figs. 3 , 4 and 5. These three graphs together

Fig. 3.

Fig. 4.

204

V. Chva'tal et al.

with C:, and the graph shown in Fig. 6 are in fact the only normalized (4,3)-graphs. Now, let us establish that: for every vertex u E H, the vertex set of H - u can be partitioned into a + 1 cliques of size w.

(3)

First, we consider the case u E G. By (2), the vertex set of G - u can be partitioned into a cliques of size w. If one of these cliques is some c k then replace this C, by C ; and C ; ; otherwise simply add C* to the a cliques. Second, we consider the case u6Z G. Now u = ak for some k. If 1 < k < w then, by the assumption, either C = Ck-lbelongs to the partition of G - uk-I or else C = ck-* belongs to the partition of G - t ) u + k - l . In either case, replacement of C by CL--2

Fig. 5 .

Fig. 6.


205

and C[-l yields the desired partition of G - ak. Finally, if k = 1 then add C:I to the partition of G - u, ; if k = w then add C L to the partition of G - u,-,. With the help of (3), proving that H is an ( a + 1, w)-graph becomes a routine matter. Let n stand for ( a + 1)w + 1 and let Y denote the n x n zero-one matrix whose columns are the incidence vectors of the ( a - 1)w 2 cliques of size w inherited by H from G and of the 2w - 1 new cliques C*, C ; , C ; , 0 s k s w -2. By this definition and by construction of H,we have JY = YJ = wJ. By (3), there is an n X n zero-one matrix X such that YX = J - I and JX = (a 1)J. As we have seen in the preceding section, these equations imply XY = J - I. In addition,

+

+

,

XJ

=

0

I

X(YJ)=

w

(J - I)J

n-1

7 -

w

J

= (a

+ 1)J.

Since each edge of H belongs to some clique of size w, the rows of X are the incidence vectors of stable sets. As in the preceding section, H had no other stable sets of size a 1. Hence H is an ( a + 1, w)-graph.

+

2. The second method

It seems that characterizing all the (a,w)-graphs may be a rather difficult problem. At the moment, we can’t even characterize those (a,w)-graphs which have circular symmetries. For these graphs, the associated matrices X , Y assume the form

x = c z’,Y = C Z’ j€B

j€A

where Z is the permutation matrix of a cycle and

(4)

IAI=a, I B ( = w . The condition XY

=J -I

reduces to

A + B ={1,2, ..., a w }

(5)

with addition modulo n = a w + 1. The graphs CZil correspond to, say, A ={1,2 ,..., w } and B ={O,w,2w ,..., ( a -l)w}. We are going to describe a more general class of solutions A, B to (4) and (5). Consequently, we shall obtain new ( a ,w )-graphs with circular symmetries. When n - 1 = m l m 2 .* mk for some integers mi greater than one, then we can consider the sets M I ,M2,. . . ,Mk defined by

-

Z=l

M.= (0, 1,. . . ,n -21. Clearly, {1,2,. . . , k } then

NOW, if niGsmi = a for some

s

V.Chvatal et al.

2ob

A

=cM,,

B=l+ZsMi

iES

satisfy (4) and (5). For example, if a

= w =4

then n - 1 = 24 and so we consider

{0,1}+{0,2}+{0,4}+{0,8}={0,1, ..., 15). Now we might choose A = {0,1) + {0,2)

B

= {0,1,2,3],

+{0,4}+{0,8}={1,5,9,13}, but instead we shall choose =1

A = {O, 1) + (0,4} = {0,1,4,5}, B = 1 + {0,2}+{0,8} = {1,3,9,11}. The latter choice yields X=Z"+Z'+Z4+Z5, Y

=z1+23+zY+z1'.

The corresponding (4,4)-graph G has vertices o0, u l , . . . ,u16 such that ui and ui are adjacent if and only if j

-

i €{2,6,7,8,9,10,11,15}

with arithmetic modulo 17. Clearly, this graph cannot be obtained by the method of the preceding section. References [I] C. Berge, Farbung von Graphen deren samtliche bnv. ungerade Kreise starr sind (Zusammenfassung), Wiss. 2. Martin Luther Univ. Halle-Wittenberg, Math. Nat. Reihe (1961) 114. 121 C. Berge, Sur une conjecture relative au probleme des codes optimaux, Commun. 13itme AssemblCe GCn. URSI, Tokyo (1962). 131 R.G. Bland, H.-C. Huang and L.E. Trotter, Jr., Graphical properties related to minimal imperfection, Discrete Math. 27 (1979) 11-22 (this volume, pp. 181-192). [4] W.G. Bridges, Jr. and H.J. Ryser, Combinatorial designs and related systems, J. Algebra 13 (1969) 432-446. [5] V. ChvBtal, On the strong perfect graph conjecture, J. Comb. Theory, Ser. B 20 (1976) 139-141. [6] C . Huang and A. Rosa, On the existence of balanced bipartite designs, Utilitas Math. 4 (1973) 55-75. [7] C . Huang, On the existence of balanced bipartite designs 11, Discrete Math. 9 (1974) 147-159. [8] C. Huang, Resolvable balanced bipartite designs, Discrete Math. 14 (1976) 319-335. 191 H.-C. Huang, Investigations on combinatorial optimization, Cornell University, O.R. Dept., Tech.Report No. 308 (August 1976). 1101 L. Lovasz, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-267 (this volume, pp. 29-42). [I 11 L. Lovisz, A characterization of perfect graphs, J. Comb. Theory, Ser. B 13 (1972) 95-98. [12] M.W.Padberg. Perfect zero-one matrices, Math. Program. 6 (1974) 180-196. [I31 A. Tucker, Critical perfect graphs and perfect 3-chromatic graphs, J. Comb. Theory, Ser. B 23 (1977) 143-149.


A CLASSIFICATION OF CERTAIN GRAPHS WITH MINIMAL IMPERFECTION PROPERTIES S.H. WHITESIDES School of Computer Science, McGill University, Montreal, Canada The family of (a, w)-graphs are of interest for several reasons. For example, any minimal counter-example to the Perfect Graph Conjecture belongs to this family. This paper accounts for all (4,3)-graphs. One of these is not obtainable by existing techniques for generating (a + 1, w)-graphs from (a, o)-graphs.

1. Introduction

A graph G is said to be perfect if for each induced subgraph G ' of G the size of the largest clique of G' is equal to the chromatic number of G'. The Perfect Graph Conjecture of Berge asserts that a graph is perfect if and only if it contains no induced subgraphs which are holes or antiholes, where a hole is a chordless cycle of odd length at least 5 , and an antihole is the complement of a hole. Lovasz [12], [13] proved a weaker conjecture of Berge: a graph is perfect if and only if its complement is perfect. The Perfect Graph Conjecture has been established for several classes of graphs, including planar graphs [16], circular arc graphs [17], KIJ-free graphs [15], [19], 3-chromatic graphs [18], and graphs with maximum degree at most 6 [71. Another way to state the Conjecture is to,say that any imperfect graph whose proper induced subgraphs are all perfect must be either a hole or an antihole. Padberg [14] showed that such a minimally imperfect, or critical, graph must be an ( a ,w)-graph, defined below. Definition. G is an (a,w)-graph if and only if (i) its vertex set has size a w + l ; (ii) its largest stable set has size a , and its largest clique has size w ; (iii) each vertex is in precisely a stable sets of size a and w cliques of size w ; (iv) each clique of size w is disjoint from precisely one stable set of a, and each stable set of size a is disjoint from precisely one clique of size w.

From now on, we denote the number aw + 1 by n. An ( a ,w)-graph is said to be normalized if each of its edges belongs to at least 207

S.H. Whitesides

208

one clique of size w. Each (a,w)-graph contains a unique normalized ( a , w ) subgraph, because removing edges which belong to no cliques of size w does not create any new stable sets of size a ((61, [ 181). The paper of Chvatal, Graham, Perold, and Whitesides [6] establishes two additional contexts in which (a,w)-graphs arise. First of all, there is a correspondence between normalized (a,w)-graphs and solutions to the system of equations

JX=XJ=aJ,

J Y = YJ=wJ, XY = J - I ,

(1)

where X and Y are matrices of 0’s and l’s, J has all entries 1, I is the identity matrix, and all these matrices are n x n. Bridges and Ryser [4]call the above matrices X and Y an ( n ,0,1) system on a, w . Second, there is a correspondence between normalized ( a ,w)-graphs and packings of the complete graph K,, by complete bipartite graphs Ka.w with each edge of K , covered exactly twice. C. Huang [9], [ 101 and C . Huang and Rosa [8] have studied such packings. The graphs denoted by CZ-’are ( a ,w)-graphs; they have vertices vo, . . . , v),,, with u, adjacent to u, whenever there is a d such that O < d < w and d = i - j or j - i (mod n ) . Holes and antiholes are of this type. In [6], methods are given for constructing (a,w)-graphs which are not of this type. The purpose of this paper is to describe all normalized (4,3)-graphs. One of these is a graph which is neither C:, nor a graph obtainable by the methods of [6]. Of course, none of these graphs is a counterexample to the Perfect Graph Conjecture. as Tucker [18] has shown the conjecture holds for graphs with w C 3. 2. Properties of ( a ,w)-graphs

We now list several well known properties of ( a ,o)-graphs which we will use frequently throughout this paper. For convenience, we will use the word clique (stable set) to refer to a clique (stable set) of maximum size only. Recall that in an (a,wkgraph, the property of being disjoint pairs off the cliques with the stable sets. We will denote the stable sets by S , , . . . ,S, and the cliques by T I , .. . ,T,, where S, n T, = 0.

Lemma 1. Zf G is an ( a ,w )-graph, then G contains exactly n = aw (of size w ) and exactly a w + 1 stable sets (of size a ) .

+ 1 cliques

Proof. This follows easily from the definition of an (a,w)-graph.

0

Lemma 2. For each vertex v in an (a,w)-graph G, the vertices of G - v are partitioned by the stable sets (cliques) which are paired with the cliques (stable sets) containing v.

Certain graphs with minimal imperfection properties

209

Proof. Let X be the matrix whose rows are the incidence vectors of the stable sets S1,.. . ,S,, and let Y be the matrix whose columns are the incidence vectors of the cliques TI,..., T,. Then we know by [6] that

JX

= XJ = aJ,

JY

=

YJ

= oJ,

and

XY

= J - I,

where J is the n x n matrix of l's, Also,

YX =X-'XYX=X-'(J-I)X =X-'JX-I=J-I. The lemma now follows from the fact that Y X = J - 1.

0

Remark 1. A consequence of Lemma 2 is that the pseudo p-critical graphs of Tucker [lS] are precisely the ( a , w)-graphs. Lemma 3. Let G be a graph, and de,finea graph M ( G ) by making the vertices of M ( G ) correspond to the cliques of G and making vertices in M ( G ) adjacent whenever the corresponding cliques intersect. If G is an (a,@)-graph, then so is M(G).

Proof. See [MI.

3. Generation of (4,3)-graphs from (3,3)-graphs We assume throughout the rest of this paper that G is a normalized (4,3)-graph whose cliques are 'triangles' T I , .. . , TI, and whose stable sets are S1,. . . , Sls,where S, n T, = P, if and only if i = j. By an i,, . . . ,ik-vertex,we mean a vertex which belongs to the stable sets S,,, . . . ,S k .By Axyr, we mean a triangle whose vertices are x, y, and z. Remark 2. We emphasize that Lemma 2 says the following: if a vertex u belongs to distinct triangles T,,, T,, and T,, then each other vertex of G is exclusively an il-vertex, an &vertex, or an &vertex. Also, it says that if distinct and Tk intersect, then there are no j,k-vertices. triangles Define a graph K ( G )from G as follows. Make the vertices of K ( G ) ,like the vertices of M ( G ) , correspond to the triangles of G. This time, however, make vertices adjacent whenever the triangles to which they correspond intersect in an edge. We first show that the maximum degree of K ( G )is at most 2. Then we use this fact to prove that K ( G ) contains a path of length at least 5 if and only if G can be generated from a (3,3)-graph by the first construction method of ChvAtal [6].

S. H.Whitesides

210

Lemma 4. The graph K ( G ) has maximum degree at most 2. Proof. If a triangle of G met three other triangles in edges, then the four triangles together would give rise to a clique of size 4 in M ( G ) . However, M ( G ) is a (4,3)-graph according to Lemma 3. 0 Lemma 5. No edge of G is in three triangles. Proof. Suppose triangle T, with vertices a, b, and c shared an edge ab with triangles T, and T,. Then since T , would intersect S, and Sk, c would be a j , k-vertex, which is ruled out b y Lemma 2. 0

Lemma 6. Suppose that H is a (3,3)-graph containing the configuration of four vertices shown in Fig. l(a) and that either {UI, 0 2 , v3} is one of three triangles partitioning H - v4 or {u2,v3,u4} is one of three triangles partitioning H - ul. Then replacing the con,figurationshown in Fig.'l(a) by the configuration shown in Fig. 1 ( b ) generates a (4,3)-graph. Proof. This is the construction of [6] specialized to the generation of (4,3)-graphs from (3,3)-graphs. 0

Lemma 7. I f K ( G ) contains a path of length at least 5, then G contains the configuration of triangles shown in Fig. 2, and (i) d is the only common neighbor of b and f ; (ii) vertices b and g can have common neighbors only if b and g are adjacent (similarly for a and f ) .

-_-._-- .-. ,.-____-------____ -... '.*,. -----r-r=1 and

IK(a1,

) N , I = l , ) N I I > l and

IKI31,

) N I l = 1 , I N z l = l and

IKI>l.

So we have to study only the following four cases:

Polynomial algorithm to recognize a Meyniel graph

23 1

= 1 and IN2/= 1, NII=O and lN21=0, = 1 and IN21G3, = 1 and IN21>3.

(a) Let us call y the vertex of K obtained by identification of y l E T ( x l )and yz E T(xz)(cf. Fig. 4). Let z, be the unique vertex of T ( x i ) \ { y ,for } i = 1,2 and let 2 ’ and z ” be respectively the exact copy of z I and z 2 in G. Suppose that there exists a vertex of G, nonadjacent to x I and a vertex of G2 nonadjacent to x 2 . Then (y, z ’ ) is a separating set for these two vertices, similarly for ( y , 2”). This is impossible in our situation for G. So we may assume that G I is a clique defined on the three vertices ( x l ,y l , zl). Therefore the amalgamation G, of G t and Gz,is isomorphic to G2,which is impossible since Gz is supposed to be a Meyniel graph. (Fig. 4 shows that the amalgamation (GI,x I ,K , ) @ ( G 2x, z . K z ),where G I and G2(x2, yz, z 2 ) are triangles, gives a graph G isomorphic to G2.) Cases (b), (c) and (d) are treated in a similar way. 0 We now state the main theorem of this paper.

Theorem 2 (Decomposition Theorem). A connected graph G = (V, E ) , which is not a basic Meyniel graph, is a Meyniel graph if and only if G can be @-properly decomposed into two Meyniel graphs GI and G2. The preceding theorem has shown that the condition is sufficient. The proof that the condition is necessary will be quite long and will constitute all of the next section. Note that in the proof of this theorem, we will use the fact that when G is decomposed into G Iand G2,GI and G2are isomorphic to certain subgraphs of G. If G is Meyniel, then GI and Gz are also Meyniel.

Fig. 4.

M. Burlet, J. Fonlupt

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4. Proof of the main theorem

To prove Theorem 2, we present a sequence of polynomial time procedures. Combining appropriately all these procedures, we get a constructive proof of Theorem 2 by an algorithm polynomial in 1 VI. For each procedure, when necessary, some remarks or special results are established before the full description of this procedure. The outputs of each procedure are followed by ‘stop’ or ‘execute procedure P,’. Instruction ‘stop’ is always accompanied either by a proof that G is not a Meyniel graph or by a proof that G satisfies Theorem 2. lnstruction ‘execute procedure P,’ means that we need to enter into procedure P, and the results obtained up to this point are the inputs of the procedure Pi. The algorithm to prove Theorem 2, which is proposed in this section, begins by verifying whether the graph G is a basic Meyniel graph. If it is not one, it tries to properly decompose G, by a special case of @ (G = (Gl,xl,K1)@(G2,xZ,K2), with K , = T ( x I )and K , = r(x2)). If such a decomposition exists, Theorem 2 is proven; if not, then there exists a hole. The algorithm detects such a hole, verifies whether this hole is even, and then examines the ‘neighbourhood’ of this hole. It may detect either of the two following configurations: a certain ‘multipartite structure’ or a certain ‘expanded basic Meyniel subgraph’ (cf. Definitions 6 and 8); if not then G is not Meyniel. The configuration detected permits the algorithm to properly decompose this graph by the operation @. For the @-decomposition found in the algorithm, i.e., G = G1@G2,GI and G, are isomorphic to certain subgraphs of G.

4.a. Is G a basic Meyniel graph?

Procedure PI. In this procedure we verify if G is a basic Meyniel graph. input : G =(V,E). (i) G is a basic Meyniel graph; stop. output : (ii) G is not a such graph; execute procedure P2. complexity : O ( I V 1).’ description: Delete S, the set ,of all the simplicia1 vertices. Detect then K, the set of all the universal vertices. Verify if G ( V \ ( K U S)) is two-connected bipartite. Verify that S is a stable set of G and x E S 3 IT(x,A)I 1. 4.b. 1s G properly decomposable by a special case of @ ?

In this subsection, we want to detect if G can be @-properly decomposed, G = ( G l , x , , K I ) @ ( G 2 , x 2 , K where 2), @ is restricted to the case where K , = I’(x,) and K 2 = T(x,). Proposition 3. Let G

= ( V ,E

) be a Meyniel graph and a disconnecting clique K


233

of G such that: there exists a universal vertex x for G(K) and G ( V \ K) has at least two connected components with more than one vertex, then there exists a simplicial disconnecting clique K1 included in K. Proof. Let K1 be a clique included in K which is minimal for this property: G( V \ K1) has at least two connected components with more than one vertex. Let GI = (Vl,E1)be the connected component of G ( V \ K1) containing x. Let G2= ( V2,E2) be another component of G ( V \ K , ) containing more than one vertex. Let y be a vertex of GZsuch that T(y, Kl) is maximal by inclusion among the sets T ( z , K 1 )where z ranges over the vertices of GZ.We claim that r(y,K ) = KI. Suppose not, then there exists z E Vzsuch that T ( z ,K1)Z T(y,K1),and also a minimal chain C(y, z ) joining y to z with all its vertices in G2.We can now find a chain (ul = y, u2, . . . , u,) whose vertices are in C(y, z ) such that:

T(u,,K1)C T(y,KI) V i Z t

and T(u,,K l ) f T(y, KI).

Let u E T(ut,K1)\T(y,Kl) and u E T(y, K,)\T(u,,Kl). Let I = max{i u E T(ui,K1), i = 1 , . . . , t } . Note that 1 < t. The induced subgraph by (ur,u l t l , . . . ,u,, u, u ) is a hole. If G is a Meyniel graph this hole is even, but adding x to this hole we obtain an odd cycle with only one chord. So T(y,K1)=K,. This proves the proposition for G2, and similarly for all the connected components of G ( V \ K,) with more than one vertex. 0

I

Procedure PZ. In this procedure we check if G contains a simplicial disconnecting clique K. A non-basic Meyniel graph G = (V, E). input : output : (i) A simplicia1 disconnecting clique K, G is properly 0decomposable, G = G1@GZ(cf. Remark 2); stop. (ii) The non-existence of such a clique K ; execute procedure P,. complexity : O ( I v I"). description: For each pair of nonadjacent vertices yl and y,, find out if K = T(yJ n T(yz)is a disconnecting clique such that yl, y2 are in different components HI,Hz of G(V \ K ) with I V(H1)I> 1 and 1 V(H*)I> 1. Remark 2. If Pz succeeds in finding a disconnecting clique K then G decomposes. More precisely, consider yi and Hi as defined in procedure Pz( i = 1,2). Let G1be an exact copy of G({yl}U K U V ( H 2 )where ) x1 corresponds to yl. Let G2be an exact copy of G({yz} U K U ( V \ ( V ( H z )U K))) where xz corresponds to y2. We have G = (GI,x l , T(xl))@(Gz,X Z , ~ ( x z ) ) .

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4.c. Detection of a hole The following proposition shows that a graph G = (V, E), which is neither a basic Meyniel graph nor decomposable by Pz, is non-triangulated, and thus contains a hole. Recall that a triangulated graph is a graph in which every cycle of length at least four has at least a chord. A triangulated graph is a Meyniel graph. Definition 5. A basic triangulated graph G = ( V ,E) is a graph wherein there exists a partition of its vertex set into a stable set S and a clique K (cf. Fig. 5). These graphs are also called split graphs. They are a particular class of basic Meyniel graphs.

Basic triangulated graph

Fig. 5.

Proposition 4. A triangulated graph G = ( V , E ) , with no simplicial disconnecting clique K , is a basic triangulated graph.

Proof. Induction on I V I . The proposition is true for I V 1 = 1,2,3. Assume that the proposition is true for I V J = n - 1. We want to prove it true for I VI = n. The graph G contains a simplicia1 vertex u. Thus G(V\{u))satisfies the hypothesis of the proposition. By the induction hypothesis, the set V \ { u } can be partitioned into a stable set S , and a clique K , . If T ( u ) CK , , the proposition is true, if not, let u be the only vertex of T ( u ) not found in K , ; the subgraph induced by ( { u } U T ( o ) \ { u } )is a clique denoted K2. We have T ( u ) C ({u1

u n u 1)-

If the vertices of V \ K 2 form a stable set, the proposition is true; otherwise, there exists a component of G ( V \ K 2 ) containing at least two vertices, thus K2\{u} is a separating set of G and G ( { u } , { u } is ) another connected component with more than one vertex of G(V\(K2\{u))).Moreover u is a universal vertex €or K2\{u}, we can apply Proposition 3. We have a contradiction. Cl


235

Procedure P3. In this procedure we detect a hole in G. G = (V, E ) after an unsuccessful application of P2. input : (i) There is an odd hole, G is not a Meyniel graph; stop. output : (ii) There is an even hole C = ( u , , u2, . . . ,u 2 k ) ; execute procedure P4. complexity : O ( I V I"). description : Choose an edge (x, y ) of G. Delete from G the edge (x, y ) and the vertices T ( x ) nT ( y ) . If x and y are linked by a chain in the resulting graph, take a minimal chain in that graph from x to y . This chain with the edge (x, y ) forms a hole (even or odd). Repeat the procedure, until we have found a hole. 4.d. Detection of an expanded cycle Proposition 5. Let C = ( u , , u 2 , . . . , u 2 k ) be an even hole in a Meyniel graph G = (V, E ) and let x E V \ V(C). One and only one of the following is true (see Fig. 6): (i) G ( V(C) U {x}) is a bipartite graph, (ii) x is a universal vertex for C, (iii) G ( V(C) U {x}) is an expanded cycle. Proof. The first two cases are mutually exclusive. Suppose that G(V(C)U{x}) is not a bipartite graph, and that x is not universal for C. Then since G is a Meyniel graph, x is adjacent to two adjacent vertices of C and is not adjacent to at least a vertex of C. By a suitable relabelling of vertices of C, we can assume that x is adjacent to u , and u2 and not adjacent to u3. Let U, be the vertex of C with the least index ( t > 3 ) adjacent to x. If t < 2k, G(x, uz, u 3 , . . . ,u , ) is a hole of G ; if this hole is odd, G is not a Meyniel graph. If this hole is even (x, u,, uz, . . . ,u,) is an odd cycle with exactly one chord. Therefore t = 2k, which means that G ( V(C) U {x}) is an expanded cycle. 0

Fig. 6.

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M. Burlet, J. Fonlupi

Proposition 6. Let G I = ( V,, E l ) be a two-connected bipartite induced subgraph of a Meyniel graph G = ( V, E ) , and x E V \ V,. One and only one of the following is true : (i) G(V, U { x } ) is a bipartite graph, (ii) x i s a uniuersal verrex for G ( V , ) , (iii) G ( V, U {x}) contains an expanded cycle.

Proof. Suppse that the first two assertions are false. An edge of El will be called a red edge, if x is adjacent to the two ends of this edge, otherwise it will be called a black edge. There exists a red edge (u, u , ) adjacent to a black edge (u, uz), since G ( V,) is a connected graph. Since G ( VI \ { u } ) is a connected graph, there exists a minimal chain in G ( V, \ { u } ) linking u , and u2. Let C, be a subchain of this one linking a vertex u ‘ to a vertex u ” such that (u, u ’ ) is red, (u, u ” ) is black, and u is not adjacent to any other vertex of C,. Then G(V(C,) U { u } ) is an even hole of (3. By Ihc preceding proposition G ( { x } U { u } U V(C,)) is an expanded cycle of G. 0 We can remark that this proof is algorithmic (in O( I V 12) . To color the edges of G, takes at most O ( I V 1)’ steps. To find the chain joining u , and u2 requires at most O ( I V )1: elementary operations. Now, to find o f and u ” requires O ( I Vl) operations. Then we have to check that G ( { x }U { u } U V(C,)) is an expanded cycle (if not, G is not a Meyniel graph). We can now describe procedure P.,.

Procedure P4. G = ( V , E ) and an even hole of G, C = ( u l r u z , . . ., u z k ) . input: One and only one of the following: output : (a) A complete multipartite induced subgraph; execute Ps. (b) A subgraph which is an expanded cycle of cardinality 5 ; execute P5. (c) A subgraph whose set of vertices is partitioned into C and K, I C I > 5 and G ( C )an expanded cycle. K is a clique containing all the universal vertices of G for G(C); execute P,. (d) G is not a Meyniel graph (a forbidden subgraph found); stop. complexity : o ( I v 13). description : V’ := V(C). Step 1. If there exists a non-universal vertex x for G ( V ’ ) ,adjacent to the vertices of an edge of G(V’), go to step 2; else, if there exists a non-universal vertex x for G ( V ’ ) ,such that Ir(x,V’)I 2 2, go to step 3; else, go to step 4. Step 2. We can apply here Proposition 6. After at most O ( 1 VI2) operations, we find either an odd hole, we have output (d), or an expanded cycle whose ) vertices ( u , ,i = 1,2,. . . ,2k + 1) are contained in V’ U { x } , ( u , , .. . , u ~an~even hole, and k k + l adjacent to u , , u2, u2k.


237

If k = 2 , we have output (b). If k > 2 and if all the universal vertices of G for the given expanded cycle induce a clique, we have output (c). If k > 2 and there exist two non-adjacent universal vertices w and w ’ , then ( u , , us,u4, w, w‘) is an expanded cycle of cardinality 5 , we have output (b). Step 3. If G ( V‘U {x}) is bipartite, V’:= V’U {x} go to step 1; else, G is not a Meyniel graph, we have output (d). Step 4. Let V, be the set of all universal vertices for G(V’). Find a minimal chain ( u , , uz,. . . ,u , ) whose vertices are in V \ ( V ’ U V,) and, among these vertices, only u , and u, are adjacent to some vertices in V’. If such a chain does not exist go to step 5; else, V‘:= V’ U ( u , , u2,. . . , u,). If G ( V’) is bipartite, go to step 1; else, G is not Meyniel, we have output (d). Step 5. At this point, as will be shown in the following remark, the universal vertices for G ( V’) do not induce a clique, or else, G is not a Meyniel graph, we have output (d). Let ( u l , u2 r . . . , u z k ) be a hole of G(V’), w and w ’ , two non-adjacent universal vertices for G ( V’). If k = 2, then G(ul,uz,u3,u4, w, w ‘ ) is a complete multipartite induced subgraph, we have output (a). If k > 2, then G ( u l , u3,u4, w, w’)is an expanded cycle, we have output (b). Complexity. Each step of this algorithm requires at most O ( I VI‘) elementary operations. We can enter at most I VI times into step 1. This procedure requires at most O ( I V )1’ operations. Remark 3. Let V, be the universal vertices for G(V’) when we enter step 5 of this algorithm. A t this point, G ( V‘) is a two-connected (bipartite) component of G ( V \ V,). Therefore, if G(V2) is a connected component of G ( V \ ( V , U V’)) there exists at most one vertex x of V’ such that V, U { x } disconnects V2 from V’\{x}. If such a vertex exists, let y be a vertex of V’ adjacent to x. Otherwise let y be any vertex of V’. Note that 1 V’I > 1, and y is universal for Vl or V, U { x } . Thus if VI is a clique and G is a Meyniel graph, Proposition 3 may be applied; this implies 1 V21= 1 otherwise procedure Pz would have found a proper decomposition of G by operation @. On the other hand, if all the connected components of G ( V \ ( V , U V’)) have exactly one vertex, G is a basic Meyniel graph which is impossible on entering procedure P4.This implies that if we enter step 5 of the preceding procedure, and if moreover V, induces a clique, G is not a Meyniel graph. 4.e. Multipartite structure Definition 6. An induced subgraph G ( V,) of a graph G = (V, E ) will be called a multipartite structure (m-structure) if V1 can be partitioned into k subclasses ( k a 2) W1, W 2 , .. . , W ksuch that G( Wi) is a connected component of G( V,),

238


1 s i s k, and at least two subclasses of the partition contain at least two vertices each (clearly this partition is unique). The subclasses with more than one vertex will be called proper subclasses. By a suitable indexing of the subclasses we shall always assume that W,, W2,. . . , W, are proper subclasses of the m-structure. We shall denote the partition of an m-structure ( W , ,W,, . . ., W,, K) where K is the union of all the non-proper subclasses.

Lemma 7. A n m-structure G(V,) of G = ( V , E ) is maximal (with respect to uertex inclusion) if and only ifthere does not exist a uertex x E V I‘ V, such that x is universal for a proper subclass of G ( V,).

Proof. Let x E V \ V,, not universal for any proper subclass of G(V,). Suppose that G(VI U { x } )is an m-structure, so there exists an edge in G ( V, U { x } )linking x to at least one vertex of any proper subclass of the m-structure G(VI). This implies that there exists exactly one connected component of G( V, U { x } ) with more than one vertex, so that G(VIU {n}) is not an m-structure. On the other hand, if x is universal, say for WI, a proper subclass of G(V,), G ( W,), will be a connected component of G ( V, U { x ) )and G((V, U {x})\ W , ) contains at least a connected component of more than one vertex. 0 We now give some results concerning maximal m-structures in Meyniel graphs.

Proposition 8. If G ( V,) is a maximal m -structure of a Meyniel graph G = ( V, E) and x E V \ V,, then x cannot be adjacent to two proper subclasses.

Proof. Let W, and W, be two proper subclasses of G ( V,). Suppose T(x, W , )# 0 and T ( x , W 2 )# 0. Since x is not universal for Wl and G (W , )is connected, there exist two vertices u,, u2 in W , such that ( u , , u2) E E, ( u , , x ) E E and (u2,x ) E E. Similarly there exist two vertices w , , w2 in W2 such that (w,, w2)6Z E, ( w l , x ) E E and ( w 2 ,x ) P E. Then (x, u,, w2,u2, w l ) is an odd cycle with only one chord ( u , , w,), which gives a contradiction. 0 Proposition 9. If G(V,) is a maximal m-structure of a Meyniel graph G = ( V ,E),whose partition is ( W , , W2, . .., W,, K ) with r > 2, there does nor exist a chain in V \ V ,joining two distinct vertices respectively adjacent to two distinct proper subclasses. Proof. If this proposition is false, let us consider the chain with the least number t of vertices contradicting this theorem. Let (u,, . . . ,u,) be such a chain. By the preceding proposition t > 1. We can suppose that


239

0, r(u’,W2)Z 0, U U l ,

WJZ

I‘(uj,Wi)=fl V l < j < t and l S i S r . Choose uoE T(u,,W,) and uc+lE T(u,,W2) and x E W,. Then (x, xo, u l , . . . , u,, ur+])is an odd cycle with exactly one chord (uo, urcl). This is impossible.

Remark 4. Let G(VI) be a subgraph of G = ( V , E ) . If G(V,) is a complete multipartite graph it is obvious that G(VI) is an m-structure. If G(V,) is an expanded cycle of cardinality 5 , G(V,) is also an m-structure. Moreover a maximal m-structure containing a complete multipartite graph or an expanded cycle with 5 vertices cannot be a graph composed of a complete bipartite graph and a clique K universal for this complete bipartite graph. Procedure Ps. A graph G = (V,E ) with the initial m-structure M given by input: procedure P4 (a complete multipartite graph or an expanded cycle with 5 vertices). (i) G is not a Meyniel graph; stop. output : (ii) A maximal m-structure with at least 3 proper subclasses. If G is @-properly decomposable then G = G1@G2;else G is not a Meyniel graph (cf. Remark 5 ) ; stop. (iii) A maximal m-structure with exactly 2 proper subclasses W , , W, such that W1 and W 2 are not both stable sets; execute Procedure P6. complexity : O( 1 V 1.)’ description: While there exists in V \ V ( M )a universal vertex x for a proper subclass of M do M:= G ( V ( M ) U {x}). If there exists in V \ V ( M )a vertex u adjacent to two proper subclasses of M, G is not a Meyniel graph, we have output (i); else we have output (ii) or (iii). Remark 5. If G(V,) is a maximal m-structure of a Meyniel graph with at least three proper subclasses, then the main theorem is true. Let ( W , , W2,.. . , W,, K) be the partition of the maximal In-structure. Delete from G the clique K and all the edges joining any vertex of Wl to any vertex of (W,, . . . , W,). By the preceding propositions W, and W , do not belong to the same connected component of this new graph. Let V’ be the connected component containing W, and V” the vertices of the other connected components. Consider the graph G I obtained by adding to an exact copy of G ( V’ U K) a new vertex xl, such that r(xl)is the entire image of WI U K. Similarly let G2be the graph obtained by

u:=,


240

adding a vertex x2 to an exact copy of G ( V ” U K), where image of ((U W, u K ). If G is a Meyniel graph we have

:=,

G

= (GI, xi, Ki)@(Gz,~

2

r(x,)is the entire

K2), ,

where we have chosen for K, the image of K in G, (i = 1,2). If we want to verify that G is Meyniel it is necessary to verify Propositions 8 and 9 (in O(l VI2)), which then permits the @-decomposition. 4.f. Expanded basic Meyniel graph

Definition 7. Let G(V,) be an induced subgraph of G = ( V , E ) . A subgraph G(V2) will be called a partitive subgraph if (i) Vz is strictly included in V, and V,l> 1, (ii) G(V2) is connected, (iii) u E V, \ V, j I‘( u. V2) = 0 or T(u, V2)= V2.

I

Definition 8 (cf. Fig. 7). An induced subgraph G ( V’) of a graph G = (V, E) will be called an expanded basic Meyniel subgraph if V’ can be partitioned into K , U , , .. . , U, ( f 3 2 ) such that: (a) u E K 3 V ‘ C { u } U T ( u ) (note that this implies that K is a clique), (b) if U, 1 2 2 , 1 s i zz f then G ( U ) is a partitive subgraph of G(V’), (c) there exists at least a partitive subgraph G ( U , )of G(V‘), 1 S i S t, (d) if we choose a vertex w, in each U,,the subgraph induced by u , , u2,.. . ,U, is a bipartite connected graph called the skeleton of G(V’), (e) for all LIE! V’, u is not adjacent, at the same time, to some vertex of U, and to some vertex of U,, where Ui and V, are represented in the skeleton by two adjacent vertices.

I

m-structure W v3

Expanded basic Meyniel graph Fig. 7.


231

The partition of G ( V’), mentioned above, will be denoted by ( K , U1,. . . . , U , ) and we will assume that U1is a partitive subgraph of G ( V’). It is important to note that an expanded basic Meyniel subgraph is not necessarily a Meyniel graph. We present here two procedures permitting us to detect, from certain P4 and P5 output configurations, an expanded basic Meyniel subgraph in a Meyniel graph.

Procedure Ps. A graph G = (V, E ) and a maximal m-structure with exactly two input : proper subclasses ( W1, W2,K ) given by P,. An expanded basic Meyniel subgraph G ( V’). If G is @-properly output : decomposable then G = GI@Gz;else G is not a Meyniel graph. (Refer to Remark 7;); stop. complexity : 0 ( 1 V 1.)’ G ( K U W1U W2) is a disguised expanded basic Meyniel subdescription: graph (cf. Remark 6). Detect the connected components of G(W,) and of G(Wz) which together with K constitute the partition of Definition 8. Remark 6. The only nontrivial point in order to prove that G ( V’) given by P, is effectively an expanded basic Mayniel subgraph of G is the condition (e) of Definition 8. But Ps assures us that this condition is satisfied when we enter P,. Procedure P7. A graph G = (V, E ) and a subgraph G(ul, u 2 , . . . , V 2 k + l ) ( k > 2 ) input : composed of an even hole ( u 1 , u 2 ,..., u z n ) and a vertex U 2 k i l adjacent to u l , uz, U Z k . Furthermore, all the universal vertices of G for this subgraph induce a clique K . (i) G is not a Meyniel graph; stop. ouput: (ii) An expanded basic Meyniel subgraph G(V‘). If G is @properly decomposable then G = G,@G,; else G is not a Meyniel graph; stop. (Refer to Remark 7.) complexity : o(i ~ 1 3 1 . description : For a vertex ui, 1 s i S 2 k , consider the subset of vertices adjacent to ui-l and ui+, (with the subscript arithmetic taken modulo 2 k ) ; consider the connected component U , containing u, induced by this subset. I f G ( K U U , ) is not an expanded basic Meyniel subgraph we have output (i); else we have output (ii).

u:El


242

Proposition 10. In a Meyniei graph G = (V, E ) , the subgraph G ( K U detected by Procedure P, is an expanded basic Meyniel subgraph.

u?:lU , )

Proof. Note that uZk+,E U1. Let u E K and u, E U,. Since ( v , , . . ., u,+ u,, a+,,.. . , v Z k ) is an even hole and u is adjacent to each v,, u is also adjacent to u,. This proves the first condition of the Definition 8. Let us show that u, E U, and u,+,E U,+, implies (u,, u , + , ) EE. Suppose it is not true: we can find u:+, and uY+, E U,+,such that (u,'+,,u,'+,)EE, (u,,u,',,)E E and (u,, u,'+,)gE. But then the cycle (ul, u 2 , . . . , or-,, u,, u,'+,,u,':~,u , , ~., . . ,u Z k )is an odd cycle containing exactly one chord. This is a contradiction, thus we have proven the second condition of the definition. Suppose there exists a vertex z P K U U, and z is adjacent to u, E U, and u , - , E L. Consider the hole C = (q,. . . ,u,-~, Y-,,u,, u , + ~. ., . ,UN). The vertex z cannot be universal for C otherwise z would be universal for the given hole ( u , , u z , ..., u Z k )and thus z would beiong to K. Proposition 5 proves that the vertex z is adjacent to three consecutive vertices of C, say u,-,, u,, v l + , . Hence (2,u , - , ) E E, otherwise the subgraph induced by ( u , , . . . ,u,-,,u,, u,+,, . . . ,u2&,z ) is an odd cycle with only one chord. Thus z E U, and we have a contradiction. 0

u?:,

Lemma 11. Let G( V , ) be a connected subgraph of a Meyniel graph G = (V, E ) with V, 1 > 1, and let z be a universal vertex for G (V,). If there exists a chain C ( x ,z ) in G(V \ V,) joining a vertex x to z where x is partially adjacent to V,, then there exists a minimal chain C,(y,z ) joining y to z where y is partially adjacent to V, and all the other vertices on C , ( y , z )are universal for VI.

I

Proof. By deleting eventually some vertices of the initial chain C(x,z), we obtain a minimal chain cl= (0,= y , uz, . . . ,u k = 2 ) whose vertices are in c(x,2 ) such that y is partially adjacent to V1and f ( o , , V,) = 0 or r ( v , , V,) = V,, V i # 1. We prove now that this chain C, is the desired chain of the lemma. If k = 2 , the lemma is proven. Suppose k > 2 and that u2 is not universal for G(VI). Let i o = min{i I'(u,, V,) = V1,uI E C , } .Consider two adjacent vertices ( w , w ' ) in V1 such that ( y , w') $ZE and ( y , w ) E E. The subgraph induced by ( w , w ' , u I , u 2 , ..,. u,) contains an odd cycle ( 35 ) with at most one chord. This is a contradiction. Suppose now that there exists a vertex of CI which is not adjacent to any vertex of G ( V,). We then can find a chain (v,, .. .,v,,,+,) Go, t 3 2 ) whose vertices are in C , ,such that only v, and ub+, are universal for G( V,). Furthermore u,-] is adjacent to at least one vertex w of V,. The subgraph induced by (uX,-,, u,, . . . ,u,,,, w ) contains an odd cycle ( 35) with at most one chord. This completes the proof. 17

1


243

uf=,

Theorem 12. Let G(V’) ( V =K U V,) be an expanded basic Meyniel subgraph of a Meyniel graph G = (V, E ) . Then there exists no chain in G ( V \ V‘) joining a vertex x to a uertex y where x is partially adjacent to U,, and y is adjacent to a vertex of U u,.

:=,

Proof. Let C(x,y ) be a chain which contradicts the theorem. The skeleton of the expanded basic Meyniel subgraph is connected, so we can concatenate the chain C(x,y ) to a suitable chain C(y,z ) to obtain a chain (vl = x , v 2 , .. . ,u, = z ) whose V,), is universal vertices belong to V \( K U U,) and where z , a vertex of for G ( U,).Moreover, by the preceding lemma we can suppose all vertices of this chain are universal for G(Ul) except u l . Let ui be the vertex of this chain with the smallest index belonging to U,). We have I < i o S r . But v+ is adjacent to ui, and also adjacent to some vertex of U1,which is impossible by definition of an expanded basic Meyniel subgraph. 0

(uf=,

(ul=,

Notation (cf. Fig. 8). Let G( V’) be an expanded basic Meyniel subgraph. Recall that we have V’ = (K U U,)) where K and U, satisfy Definition 8. For the remainder of this section, let us define: (i) is the union of U , and the set of vertices belonging to some connected component of G ( V \ ( K U U,)) such that this component contains at least a subgraph G(U,) i # 1, (ii) is the set of elements of V \ contained in some connected component of G(V\(K U U , ) )such that this component is adjacent to U1, (iii) K = { u E K ~ ( u P2) , z 01.

Cut=,

e,

vl

v2

1

M. Burlet, J. Fonlupt

244

Theorem 13. Let G ( V‘) be an expanded basic Meyniel subgraph, and V,, Vz,K as defined aboue. Then: (a) U ,is a partitive subgraph of G ( (b) each uertex of K is adjacent to all the vertices of universal for U , .

el),

v,

Proof. (a) This part is a reformulation of the preceding theorem. (b) Let z I be a vertex of linking z, and zz such that

K. Let z z E UI be such that there

exists a chain

e (*1

0 # ( V ( C ) \ { Z l , 2 2 ~ ) CQz.

Let C be the shortest among those chain satisfying (*). Let z be a universal vertex of G ( V’\K) for UI.Note that z is adjacent to 2 , and zz and to no other vertex of C. Therefore if I V(C) fl > 1 the subgraph induced by (V(C) U { z } ) contains either an odd hole 5 5 or an odd cycle 3 5 with only one chord. So we can assume that V(C) n ={z3}. Suppose that part (b) of the theorem is false. Consider a shortest chain C ( y l ,y z ) joining y , E to some y2 E V‘\(K U U , )where y l is universal for U , and not adjacent to a vertex z, E I% Let zz and z3 be defined relatively to z1 as stated above. Since the skeleton of G(V’) is connected, consider a chain C ( y Z r y 3joining ) y 2 to y?, whose vertices are in V‘\(K U U , ) and where y l is universal for U,. Concatenating C ( y l , y 2 )and C ( y z , y 3 )we obtain a chain CI= ( u , , . . . ,u,, . . . ,u I ) with u , = y , , u, = y z and of = y 3 . We can suppose that C(y,,y 3 ) is a minimal chain, which obviously implies that CI itself is a minimal chain. If r = 2 , u, cannot be adjacent to Ul (cf. Definition 8). Therefore G(zl,z2,z3,ul,u z ) is an odd cycle with only one chord; this is impossible. Suppose r > 2 and uz not universal for UI. Let u, be the vertex of Cl with the smallest index such that (ub,z r ) EE; note josr. The vertices u, with 1 < i S j,, cannot be adjacent to UI by the way we chose the chain C ( y l , y 2 ) . If u,, is universal for U , , the subgraph induced by (zl,1 2 , u I ,u z , ... , u,) is not Meyniel. If u, is not universal for U , ,the subgraph induced by (z3, zzrz,, u I , uz, . . . , uk,)is not Meyniel. Thus uz has to be universal for U , .This enables us to show that all the vertices of the chain ( u , , . . . ,u , ) are universal for U , . If this were not the case, we could find two vertices of CI,ub and u, UO> io + 1 b 3) such that u,, D,, u2,. ..,u,, are universal for U , but not u,,,, . . . ,u,-,. The subgraph induced by the vertices (zz, u,-,,. . . ,u,) is not Meyniel, which leads to a contradiction. Therefore u,-, (which V’) is universal for U , and adjacent to U, (which P V’ and is universal for U , )and this is impossible by definition of an expanded basic Meyniel graph. This completes the proof.

v21

vz

v,

Remark 7 (see Fig. 9). Take an exact copy of G ( V \ e2)and let GI be this copy where the set of vertices corresponding to those of U , reduced to a single vertex,


245

which we will call x,. Let K1 be the clique of GI which is the exact copy of K. Take an exact copy of G ( U UIU I?) and let Gz be this copy to which we add a new vertex xz, adjacent only to all the vertices corresponding to those of K U U I . Let K 2 be the clique of Gz which is the exact copy of I?. If G is Meyniel, G = ( G I , x , , K l ) ~ ( G 2 , x Z , KIfZ )we . want to verify that G is Meyniel, it is necessary to verify that UIis a partitive subgraph of G( and that every vertex of I? is adjacent to every universal vertex of for G ( V , )(in O ( 1 V IZ) , which would then permit the @-proper decomposition.

v2

el

vl)

5. Polynomial recognition algorithm of a Meyniel graph and applications 5.a. Polynomial recognition algorithm

We are going to give the description of a polynomial recognition algorithm of a Meyniel graph. Let G o = ( V " , E " ) be the graph to be recognized. The preceding algorithm enables us in polynomial time to detect for G o one and only one of the following situations: (i) G o is not a Meyniel graph, (ii) G" is a basic Meyniel graph, (iii) Go is not a basic Meyniel graph, but there exists a @-proper decomposition into two graphs G ' = (VI, E l ) and GZ= (V', EZ)and, moreover, G o is a Meyniel graph if and only if G ' and GZare Meyniel graphs. The idea of the recognition algorithm presented below will be to apply the algorithm of the preceding section to the graph G o and, inductively, to each new graph possibly obtained by this algorithm.

M.Burlei. J. Fonlupi

216

Definition 9. Let Ce be the family of graphs obtained by the recognition algorithm applied to G".A t the start of the algorithm, the family Ce contains only one element, G". When the algorithm stops the graphs of Ce will have been denoted G' ( i = O , l , . . . , ICeI-l), where G ' = ( V ' , E ' ) . The recognition algorithm Begin

%:=(GOj. k:=O. while some graphs in % have not yet been examined by the decomposition algorithm Begin Apply the algorithm of Section 4 on this graph. If this graph is not Meyniel, GO is not Meyniel; stop. If this graph is a basic Meyniel graph go to continue. If this graph is decomposed into two new graphs, call them G k + 'G , ktZ. %:= 92 U { G k + U ' } {Gk+'}. k:= k +2. continue: end

end

Since the algorithm permits only proper decompositions, it is obvious that it will stop. Definition 10. Let k(G")be the cardinality of the largest family % which can be obtained by applying the recognition algorithm on Go.The graphs contained in this family are indexed from 0 to k(Go)- 1. If a graph G' of a family 9 is decomposed by a @-decomposition into G' and G', G' and G h are called successors of G'. Let K' be a clique of maximum cardinality of G', and let r, = 1 V ' l - / K ' l . To prove that the recognition algorithm is polynomial we only have to show that k(G")is of polynomial order in I V o J . Theorem 14. If G' and G kare successors of G ' then (a) I V'I < I V'I and I V kI < 1 V ' 1, 1 K J I K' I and 1 K kI 1 K ' 1, r, c r, and rk d r,. (b) r, + rk S r, + 1.

I

Proof. (a) This part is a direct consequence of the definition of a @-proper decomposition.


247

(b) We shall prove part (b) for Go, G ' and G2. Let Go= (GI, xl, K1)@(G', x2, K2).Let K be the clique of G" obtained by the identification of K , and K2of the operation Qi. Let V? (resp. V:) be the subset of vertices of G o corresponding to the vertices V1\({xl}U Kl) (resp. V'\({x,} U K2))of G ' (resp. G'). Let N': (resp. N:) be the subset of vertices of Gocorresponding to the vertices T(xl)\K1(resp. r ( x 2 ) \ K 2 )of GI (resp. G2). Note that ro= I V?l+ I GI+IKI - IK'I. First case : V? n K" # 0 and @ n K" # 0. n KO lie respectively By definition of the amalgam operation, V? n KO and in NY and N:. Again, by definition of @, KO = (KO n N?)U (KO n Nz) U K. The subgraph of G ' induced by {xl}, K I and the subset of vertices of G ' corresponding to NY n KO is a clique of cardinality: I K 1 + 1+ 1 N': n KO 1 . Therefore r, s I V ( G ' ) l - ( l K l + 1+ 1N':n K o l ) = I V?l- I N? n K"1. Similarly r z S I V':l- I Ny n K" 1. As ro = 1 fll+ 1 El- IN': n KO1 - IN; n K"I, we get

rl + r2 s ro. Second case: V; r) K" = 0. (The case V? n KO = 0 is similar). There exists in GI a clique of cardinality I K'I. As I V(G')I = I K we have rl

S

I + I V?l+ 1,

1 V':l+ IKI + 1 - IK"I.

Moreover ({x2} U K 2 ) induces a clique of cardinality IK r z d ) V ( G 2 ) ) - ) K J - 1 = I V ~ Iand , we get

rl+r2 my' > TL', which implies that ( u i , uk)E E This is only half of the story; we actually have the following result of Pneuli, Lempel and Even [20]. Theorem 7.1. A n undirected graph G is a permutation graph if and only if G and G are comparability graphs.

Theorem 7.1 suggests an algorithm for recognizing permutation graphs, namely, applying the transitive orientation algorithm to the graph and to its complement. If we succeed in finding transitive orientations, then the graph is a permutation graph. To find a suitable permutation we can follow the construction procedure in the proof of the theorem, which can be found in Golumbic [13]. The entire method requires O ( n 3 )time and O ( n Z )space. Permutation graphs are useful in a number of applications (Even, Pnueli and Lempel [ 5 ] ,Tarjan [25],Golumbic [ 131). Of particular interest in this context is the following very efficient coloring algorithm for G [ T ] . Algorithm 7.1. Coloring a Permutation Graph. Input: A permutation T = [T,, T ~ ,. ..,T"]of the numbers {1,2,. . . ,n } . Output: A coloring of the vertices G [ T ]and the chromatic number y, of G [ T ] . Method: The vertices of G [ T ]are assigned colors in the order m,7r2,. . . , r n , although the graph itself is never actually calculated. A counter k will keep track of the total number of colors used so far, and an array LAST(c) will contain the number of the vertex which was the last to receive color c. During the jth time

M.C. Golumbic

318

through the loop we color T, with the smallest color q satisfying rj2 LAST(q). The entire algorithm is as follows: procedure

k t o ; for i + 1 to n do LAST(i)+O; for j + l to n do m t rnin{q [ nj 3 LAST(q)}; COLOR(rj)+m; LAST( m )t rj; k +max{k, m}; end loop

1. initialize: 2. loop: 3. 4. 5. 6. 7.

end

X+k; end

Example. Let us illustrate Algorithm 7.1 on the permutation T = [4,1,3,5,2]. After the initializations in line 1 the fotlowing assignments will be made in the loop: j +Z rn +2

I +3

jt 4

j--S

rn +2

m+l

m -3

COLOR(l)+2 LAST(2)tl k +2

COLOR(3)+2 LAST(2)+3 k +2

COLOR(S)+1 LAST(l)+5 k6 2

COLOR(2)+3 LAST(3)+2 k +3

j-I

m t l COLOR(4)c 1 LAST(l)t4 k-1

Thus the chromatic number of G [ r ]is 3 and a 3-coloring has been assigned. The complexity of Algorithm 7.1 is O ( n l o g x ) if line 3 is implemented using binary search. A proof of the correctness of this algorithm can be found in Golumbic [13]. Algorithm 7.1 can be used to color any permutation graph G in O(rz log n ) time provided we are given the permutation r and the isomorphism G + G [ r ] .If we do not have r,then we should use Algorithm 5.4.

8. Interval graphs An undirected graph G is called an interval graph if its vertices can be put into one-to-one correspondence with a set of intervals 9 of a linearly ordered set (like the real line) such that two vertices are connected by an edge of G if and only if their corresponding intervals have nonempty intersection. We call 9 an interoal representation for G. (It is unimportant whether we use open intervals or closed intervals; the resulting class of graphs will be the same.) The following characterization of interval graphs is due to Gilrnore and Hoffman [lo].

Algorithmic aspects of perfect graphs

319

Theorem 8.1. An undirected graph G is an interval graph if and only if G is a triangulated graph and its complement G is a comparability graph.

The coloring, clique, stable set, and clique cover problems can be solved in polynomial time for interval graphs by using the algorithms of Sections 4 or 5 , and a recognition algorithm could be obtained by combining the algorithms for triangulated graphs and comparability graphs. However, the recognition algorithm presented in Booth and Lueker [3] is asymptotically more efficient. They have shown that a data structure called a PQ-tree can be used to obtain a linear algorithm. Interval graphs have become particularly useful mathematical structures for modeling real world problems. The line, on which the intervals rest, may represent anything that is normally regarded as one-dimensional. The linearity may be due to physical restriction such as blemishes on a microorganism, speed traps on a highway, or files in sequential storage in a computer. It may arise from time dependencies as in the case of the life span of persons or cars, or jobs on a fixed time schedule. A cost function may be the reason as with the approximate worth of some fine wines or the potential for growth of a portfolio of securities. The task to be performed on an interval graph will vary from problem to problem. If what is required is to find a coloring or a maximum weighted stable set or a large clique, then fast algorithms are available. If a Hamiltonian circuit must be found, then there are no known efficient algorithms (unless the graph has more structure than just being an interval graph). Also, the speed with which such a problem can be solved will depend partially on whether we are given simply the interval graph G, or, in addition, an interval representation of G. We have already seen one application of interval graphs in the opening paragraph of this article. The interested reader is referred to Roberts [21], [22] and Golumbic [13] for numerous other applications. We will discuss here a recent application of interval graphs to optimal macro substitutions suggested by Golumbic, Goss and Dewar [ 151. The compiler or interpreter for a microcomputer system may be regarded as a byte sequence which resides in main memory. Due to restrictions on the size of main memory, it is desirable to compact this byte sequence. One technique is to define a set of macro substitutions which allow occurrences of specified byte subsequences to be replaced by single bytes. The subsequences are restored dynamically at run time by use of an associated table. Fig. 8.1 shows a sequence of hexidecimal digits of length 36. Since the digits E and F do not appear, they may be used to indicate macros. Choosing E = 6A2 and F = 43B96 the original sequence may be reduced to length 20. Notice that when two macros overlap, only one can be replaced. This overlapping phenomenon, therefore, restricts how the macro table may be applied.

M.C. Golumbic

320

~

Original Sequence: 6&C43B%OD60661C78-243B96&3C&25 Macro Table: E = 6A2

F = 43B96

T \

OVERLAP

Abbreviated Sequence: ECFOD6OElC78EFA23CE5

Fig. 8.1. Macro substitution.

The problem to be solved is to choose an optimal set of macro substitutions and an order for performing the substitutions which minimizes the total length of the byte sequence and associated table. Formally we require the following. Input: Output:

A byte sequence B of length n.

A set of m macros each of length s k and an order for performing the substitutions such that the total length of the abbreviated sequence and macro table is minimized.

The reason for specifying a bound on the length of the macros is that in practice we may want them to be very short compared to the length of the original sequence. Notice that there are actually two aspects to the problem: (1) choosing a macro set, and (2) using the macro set optimally. Let B = ( b , ,bz,. . . ,b,) be a sequence of bytes and let k be a fixed constant. The length of B is denoted by I B I = n. A subsequence ( h , .. . ,b,) of B is denoted by B [ i . j ] . Clearly, l B [ i , J ] l= J - i + 1. The weighred interval graph G = (V, E, w ) that we will associate with B is defined as follows: The vertex set V consists of all intervals [ i , j ] satisfying 1 s J - i k - 1 ; two vertices u = [ i , J ] and u = [ i ' , j ' ] are connected by an edge iff they intersect, i.e., either i f d j d j' or i S J ' s j ; the weight w ( u ) of a vertex u = [ i , J ] is equal to j - i which represents the number of bytes that would be saved by replacing B [ i , j ] by a single byte. I t is easy to see that the number of vertices of G is slightly less than kn and the number of edges is less than but on the order of k'n. Furthermore, the graph does not actually have to be calculated and stored since any query about adjacency of vertices can be answered by a simple comparison of the indices of their corresponding subsequences. Let M be a subset of V and let

1

B [ M ] = { B [ i , j ] [ i . J ]E M ) . We may think of B [ M ] as the macro table generated by M. To perform the


32 I

macro substitutions we would find all occurrences of these macros and then choose a subset of the occurrences, no two of which intersect, to be abbreviated. Such a subset corresponds precisely to a stable set of the interval graph G. (Notice that this model does not permit embedding one macro in another macro.) Moreover, t o make the abbreviated sequence as short as possible, we would like a stable set whose weight is maximum. (The weight of a subset of vertices is the sum of the weights of its members.) This method is summarized in Fig. 8.2. procedure SUBSTITUTION(A4): C ( M ) t { [ i j, ] E V B [ i ,j ] = B [i ‘ , j ’ ] for some [ i ’ , j ’ ] E M } ;

I

X ( M ) t M A X I M U M WEIGHTED STABLE SET OF THE INDUCED SUBGRAPH GC(M); SAVINGS(M)+

c

“EX(M)

w(u)-

c

w(u);

“EM

end Fig. 8.2. Finding an optimal macro substitution for a given set of macros,

The set C ( M ) consists of all intervals representing candidate subsequences which may be replaced using the macro table B [ M ] . Of these candidates only the subsequences represented by X ( M ) will be replaced. The SAVINGS is calculated by summing the savings obtained for each macro substitution and subtracting the cost of storing the macro table. Using SUBSTITUTION we obtain the following algorithm which gives an optimal solution to the general problem. Algorithm 8.1. V such that IM I = rn do loop: for all M call SUBSTITUTION(M); end loop return the M and X ( M ) whose SAVINGS(M) is maximum; T h e number of passes through the loop in Algorithm 8.1 is on the order of (?:) since G has O ( k n ) vertices. (In practice, some of the subsets M may be ruled out due to other criteria, for example, by requiring that macros begin with certain designated bytes. This would lower the number of passes.) The complexity of SUBSTITUTION depends on how efficiently we are able to find C ( M ) and X ( M ) for a given M . Using a modification of the deterministic pattern matching algorithm of Morris and Pratt [19’], C ( M ) can be calculated in O(rn(k n ) ) time. See also Aho, Hopcroft and Ullman [l], Chapter 9. Since a maximum stable set of a n interval graph G = (V, E ) may be found in time O(l VI + IE I), X ( M ) can be calculated in O ( k ’ n ) time. Hence, we conclude

+

322

M.C. Golumbic

that the worst case complexity of SUBSTITUTION is O ( m ( k + n ) + k ' n ) and the worst case complexity of Algorithm 8.1 is

which is, in terms of the length of the input sequence, a polynomial whose degree depends on the constant m. Notice that our model has not allowed the embedding of macros in other macros. A reason for this could be that it is impractical to implement the stack nccessary to allow embedding. In some applications one may choose to allow embedding. If this is the case, a similar model can be designed which uses overlap graphs rather than interval graphs. An overlap graph is the same as an interval graph in which there are no edges between pairs of vertices whose corresponding intervals have one properly contained in the other. Our Algorithm 8.1 and SUBSTITUTION will also be optimal using the overlap graph model. Their respective complexities, in this case, will each be raised by one power of kn. This follows from the fact that a maximum weighted stable set of an overlap graph G = (V, E ) can be calculated in O ( I V 1. I E I) time (see Gavril [Y] and Golumbic [13], Chapter 1 I). The problem of macro substitution was recently applied to MICRO SPITBOL for an Incoterm SPD20/40 supporting 64K of main memory. The byte sequence for MICRO SPITBOL required 23,110 bytes of storage. There were 176 unused opcodes which were designated to represent macros. That is, n = 23110 and m = 176 and we set k =20. Since the time complexity of Algorithm 8.1 would be high for this application, an effective technique for finding a near optimal solution was needed. A combination of heuristics and SUBSTITUTE reduced the size of the sequence to 17.920 bytes and produced a macro table of 962 bytes. This represents a saving of 4,228 bytes of main storage, a saving of 20%. It should be pointed our that an increased cost of obtaining a very good macro substitution may be justified by the fact that this is done only once per compiler and machine and the result presumably will be used many, many times.

References [ I I A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, Reading, MA, 1976). [2] C. Berge, Graph and Hypergraphs (North-Holland, Amsterdam, 1973). [ 3 ] K.S. Booth and G.S. Lueker, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, J . Comput. Systems Sci. 13 (1976) 335-379. 141 G . A . Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71-76.


323

[5] S. Even, A. Pnueli and A. Lempel, Permutation graphs and transitive graphs, J. Assoc. Comput. Mach. 19 (1972) 4 W 1 0 . [6] S. Foldes and P.L. Hammer, Split graphs, in: F. Hoffman et al., eds., Proc. 8th Southeastern Conf. on Combinatorics, Graph Theory and Computing (Congressus Numerantiurn XIX, Utilitas Math., Winnipeg, 1977) 311-315. [7] D.R. Fulkerson and O.A. Gross, Incidence matrices and interval graphs, Pacific J. Math. 15 (1965) 835-855. IS] F. Gavril, Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph, SIAM J. Comput. 1 (1972) 180-187. [9] F. Gavril, Algorithms for a maximum clique and a minimum independent set of a circle graph, Networks 3 (1973) 261-273. [The class of circle graphs is equivalent to the class of overlap graphs.] [lo] P.C. Gilmore and A.J. Hoffman, A characterization of comparability graphs and of interval graphs, Canad. J. Math. 16 (1964) 539-548. [ l l ] M.C. Golumbic, Comparability graphs and a new matroid, J. Comb. Theory, Ser. B 22 (1977) 68-90, [ 121 M.C. Golumbic, The complexity of comparability graph recognition and coloring, Computing 18 (1977) 199-208. 1131 . . M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). 1141 M.C. Golumbic, A remark on the NP-completeness of the threshold dimension problem, Bell Laboratories Technical Memorandum (1982). [15] M.C. Golumbic, C.F. Goss and R.B.K. Dewar, Macro substitutions in MICRO SPITBOL - A combinatorial analysis, in: F. Hoffman et al., eds., Proc. 11th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Congressus Numerantium (Utilitas Math., Winnipeg, 1980) 485-495. [16] M. Grotschel, L. Lovrisz and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981) 169-197. [17] P.L. Hammer and B. Simeone, The splittance of a graph, Combinatorica 1 (1982) 275-284. [18] C.G. Lekkerkerker and J. Ch. Boland, Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51 (1962) 45-64. [19] J.H. Morris and V. R. Pratt, A linear pattern matching algorithm (Tech. Report No. 40, Computing Center, University of California, Berkeley, Calif., 1970). [20] A. Pnueli, A. Lempel and S. Even, Transitive orientation of graphs and identification of permutation graphs, Canad. J. Math. 23 (1971) 160-175. (211 F.S. Roberts, Discrete Mathematical Models, with Applications to Social, Biological and Environmental Problems (Prentice-Hall, Englewood Cliffs, New Jersey, 1976). [22] F.S. Roberts, Graph Theory and Its Application to Problems of Society, NSF-CBMS Monograph No. 29 (SIAM Publ., Philadelphia, Pa., 1978). [23] D.J. Rose, R.E. Tarjan and G.S. Lueker, Algorithmic aspects of vertex elimination on graphs, SIAM J. Comput. 5 (1976) 266-283. [24] L.N. Shevrin and N.D. Filippov, Partially ordered sets and their comparability graphs, Siberian Math. J. 11 (1970) 497-509. [25] R.E. Tarjan, Sorting using networks of queues and stacks, J. Assoc. Comput. Mach. 19 (1972) 341-346. [26] R.E. Tarjan, Maximum cardinality search and chordal graphs (Stanford Univ. Lecture Notes CS 259, unpublished, 1976). [27] W.T. Trotter, Jr., J.I. Moore and D.P. Sumner, The dimension of a comparability graph, Proc. Amer. Math. SOC.60 (1976) 35-38.

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POLYNOMIAL ALGORITHMS FOR PERFECT GRAPHS M. GROTSCHEL Institut fur Mathematik, Universifiit Augsburg, Augshurg, W. Germany

L. LOVASZ Institute of Mathematics, Eiitviis Lorand University, H - 1088 Budapest, Hungary

A. SCHRIJVER Department of Econometrics, Tilburg University, Tilburg, The Netherlands We show that the weighted versions of the stable set problem, the clique problem. the coloring problem and the clique covering problem are solvable in polynomial time for perfect graphs. Our algorithms are based on the ellipsoid method and a polynomial time separation algorithm for a certain class of positive semidefinite matrices related to Lovasz’s bound 8 ( G ) on the Shannon capacity of a graph. We show that 9 ( G ) can be computed in polynomial time for all graphs G and also give a new characterization of perfect graphs in terms of this number 9(G). In addition we prove that the problem of verifying that a graph is imperfect is in NP. Moreover, we show that the computation of the stability number and the fractional stability number of a graph are unrelated with respect to hardness (if P # N P ) .

1. Introduction and notation It is well known that the stable set problem, the clique problem, the chromatic number problem and the clique cover problem are NP-complete problems for general graphs, cf. [3]. The purpose of this paper is to show that these problems, and even their weighted versions, are solvable in polynomial time for perfect graphs. The algorithms presented here are based on the ellipsoid method (cf. [XI, [2], [4])and on a computationally tractable characterization of the number 8 ( G ) introduced by Lovasz 1101in connection with the Shannon capacity of a graph. In the remaining part of this section we shall introduce our notation and state the problems we shall investigate. The second section gives a brief review of the ellipsoid method and some properties of this method which are important for our purposes. In Section 3 we show that the stable set problem is unrelated to the fractional stable set problem with respect to hardness for general graphs. The Shannon capacity and the numbers 6 ( G ) , &,(G), which are important for the design of our algorithms, are treated in Section 4, and a polynomial separation algorithm for a certain class of positive semidefinite matrices related to 6 ( G ) is presented in Section 5. This algorithm is utilized together with the ellipsoid 325

M.Grotschel et a/.

326

method in Sections 6 and 7 to obtain polynomial time algorithms for the weighted versions of the stable set problem, clique problem, coloring problem and clique cover problem on perfect graphs. All graphs in this paper are finite and undirected. Since loops and multiple edges do not play a role for the concepts we consider, we assume that all graphs are without such edges, i.e., are simple. A graph is denoted by G = (V(G), E ( G ) )where V(G) (or just V) is the vertex set and E ( G )(or just E ) is the edge set of G. An edge connecting two vertices i and j is denoted by ij, and we say that two vertices are adjacent if they are equal or connected by an edge. The complementary graph of a graph G is defined as the graph G with V(G) = V(G) and in which two different vertices are adjacent if and only if they are nonadjacent in G. A stable set of a graph G is a set of vertices W C V(G) such that any two vertices of W are nonadjacent in G, and a clique of G is a set of vertices C c V(G) such that any two vertices of C are adjacent in G. The maximum cardinality of a stable set in G is called the stability number of G and is denoted by a ( G ) .The maximum cardinality of a clique in G is called the clique number of G and is denoted by w ( G ) . Clearly, a stable set of G is a clique of G, and vice versa, thus a ( G )= w ( G ) and o ( G )= a(G)hold. A k-coloration of G is a partition of V(G) into k stable sets of G, and the least integer k for which G admits a k-coloration is called the chromatic number of G, denoted by x ( G ) . A k-clique couer of G is a partition of V(G) into k cliques of G, and the least k for which G admits a k-clique cover is called the clique couer number of G which is denoted by p ( G ) . By definition, every k-coloration of G is a k-clique cover of and vice versa, which implies x ( G ) = p ( G ) and p ( G )= x(G). The problem of finding the stability number (clique number, chromatic number, clique cover number) of a graph is called the srable set (clique, coloring, clique couer) problem. These four problems have natural weighted versions. Given a graph G = (V, E ) and a ‘weight’ w, E E , for all u E V (H, is the set of positive integers), then the weighted srable set problem (weighted clique problem) is to find a stable set W (a clique C) of G such that the sum of the weights of the vertices in W (in C) is as large as possible. The weighted coloring problem (weighted clique problem) is the following: find stable sets W,, Wz,.. ., W, (cliques C1,Cz,.. . ,C,) and positive integers y l , y z , . . . ,y , such that for all u E V, y , 3 w, y , 2 w , ) holds, and such that y , is as small as possible. The optimum values of these four problems are denoted by a, (G), ow(G), xW(G), pw( G ) and are called the weighted stability, weighred clique, weighted chromatic, weighred clique couer number. It is obvious that for any graph G, a ( G ) S p ( G ) and w ( G ) S x ( G )hold. A graph G is called perfect if

c,

c,,,

(c,,,

c:=,

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327

a ( G [W ] )= p ( G [W ] ) for all W C V ( G ) where G [W ] denotes the subgraph of G induced by the vertex set W C V ( G ) . Lovasz [9] has shown the so-called perfect graph theorem, namely, that a graph G is perfect if and only if its complement G is perfect. So the perfect graph theorem is equivalent to the following: a graph G is perfect if and only if w ( G [ W ] ) = x ( G [ W ] )for all W C V ( G ) . Due to the perfect graph theorem it suffices to design polynomial time algorithms only for the weighted stable set and the weighted clique cover problem in order to obtain polynomial time algorithms for all the four problems described above on perfect graphs. Namely, suppose we have a polynomial time algorithm for the weighted stable set problem on perfect graphs and we want to find the maximum weighted clique in a perfect graph G. Then, obviously, the set of maximum weighted cliques of G equals the set of maximum weighted stable sets of G. Since by the perfect graph theorem G is perfect, we can apply our polynomial time algorithm to calculate a maximum weighted stable set in G and thereby obtain a maximum weighted clique in G. Similarly, if we have a polynomial time algorithm for the weighted clique cover problem in perfect graphs we can obtain a minimum weighted coloring of a perfect graph G by applying our polynomial time algorithm to the (perfect) complementary graph G. Therefore, we shall concentrate in the sequel on designing polynomial time algorithms for the weighted stable set and clique cover problem on perfect graphs, keeping in mind that these also yield polynomial time algorithms for the weighted clique and coloring problem on perfect graphs. There are various classes of graphs known for which the weighted versions of the stable set, the clique, the coloring or the clique cover problem can be solved in polynomial time. For a survey of such results see [3]. These classes of graphs include several classes of perfect graphs, e.g., bipartite, triangulated and comparability graphs as well as line graphs of bipartite graphs. Recently, Hsu [6] has shown that the coloring problem, and Hsu and Nemhauser [7] have shown that the clique and clique cover problem, are solvable in polynomial time for claw-free perfect graphs.

2. The ellipsoid method

Based on an algorithm due to Shor [13], Khachiyan [S] recently devised a method which solves linear programming problems in polynomial time; for nrnofs. see 121. This so-called ellipsoid method can be used to derive the

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polynomial solvability of a more general class of problems, in particular to obtain a powerful tool for solving combinatorial optimization problems as was described by Grotschel et al. [4].In this section we give a brief survey of this method and state those theorems of Grotschel et al. [4]which are of interest for the design of polynomial time algorithms on perfect graphs. A conuex body is a closed, bounded, fully dimensional, and convex subset of R", n 3 2. More precisely, if we speak of a convex body K we always assume that the following information is known: the integer n 3 2 with K C R", two rational numbers 0 < r S R, and a vector a. E K such that

S(a,, r ) C K 5 S(ao,R ) , where S(a,,,s ) = {x E R" IIIx - aoIIS s } (11. ( 1 is the euclidean norm), denotes ball with center ao and radius s. Therefore, we also denote a convex body by quintuple (K; n, ao,r, R ) where we assume that n 5 2, a. E Q", 0< r G R given explicitly. The following two problems are of particular interest and - as we shall later - polynomially related. Assume that a convex body ( K ; n,a,,,r, R ) is given.

the the are see

(2.1) Optimization Problem. Given a vector c E QD" and a rational number E > 0, find a vector y E 6)" such that d(y, K) =z E and cTx =z cTy + E for all x E K (i.e. y is almost in K and almost maximizes cTx on K). (2.2) Separation Problem. Given a vector y E Q" and a rational number S > 0,

conclude with one of the following: (i) asserting that d ( y , K )4 6 (i.e., y is almost in K); or (ii) finding a vector c E Q" such that IIc 11 2 1 and for every x E K, cTx 6 cTy+ S (i.e. finding an almost separating hyperplane). (Here d ( . ,.)denotes the distance function, i.e., d ( x , y) = \ \ x - y 11 and d(y, K ) = inf{d(x, y ) Jx E K ) . ) The method of Yudin and Nemirovskii [14], section 4.5, would enable us to show that the following third problem is also polynomially related to problems (2.1) and (2.2) above:

(2.2') Feasibility Problem. Given a vector y E Q" and a rational number S > 0. conclude with one of the following: (i) asserting that d ( y ,K) S 6, or (ii) asserting that d (y, R"\ K ) s 6. Clearly this problem is easier than the separation problem. However, in the


329

applications in this paper, we shall obtain almost separating hyperplanes automatically. To speak of a polynomial time algorithm for a convex body K we have to specify how we measure the input length of K. Whenever something is encoded we assume that the (usual) binary encoding is used. A rational number is encoded by encoding the numerator and the denominator. If x E Q" (Q is the set of rational numbers) then Ilx denotes the maximum of the absolute values of the integers appearing as numerator or denominator in the coefficients of x. In other words, to encode x at least log IIx 1 1 + n places are necessary. (In this paper all logarithms have base two.) For a convex body ( K ;n, a,,, r, R ) we assume that the parameters n, allE Q", r E 4p and R E Q are coded. If X is a class of convex bodies then the input of the optimization (or separation) problem for Yt is the code of some member ( K ; n, acl,r, R ) E X, of a vector c E Q" and of a rational number F > 0 (of a vector y E q" and a rational number 6 > 0). The length or size of the input is the length of this (binary) encoding. Thus, the length of the input is at least

)I

+ 1%

It r

1 1 3

It It= + log It Y 11%

+ log R

where y = E or y = 6. An algorithm to solve the optimization (separation) problem for the class Yt is called polynomial if its running time is bounded by some polynomial of the size of the input. (2.3) The ellipsoid method. Given a convex body ( K ;n, all,r, R ) , a linear objective function cTx with IIc ( 1 3 1 and a number E > 0 (the required accuracy). We assume that there is a subroutine SEP(K, y, 6 ) which for the given convex body K, a vector y E Q" and a rational 6 > 0 either concludes that y E S ( K ,6 ) = {x E R" d ( x , K ) s 8 ) or yields a vector d E Q" such that dTx S dTy + 6 for all x E K, i.e. SEP solves the separation problem for K. We first define the following numbers:

1

, (2.3.1) N:=4n2[ l o g m ] r&

(2.3.2)

24-N a:=- R300n

12Gl

(2.3.3) p : = 5 N (log 7 and then proceed as follows:

(2.3.4) Set x , ~= a. (center of the first ellipsoid), A,i:= R'I, (I,,is the ( n , n)-identity matrix);

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(2.3.5) for k = 0 to N - 1 do; (2.3.6) Run the subroutine SEP(K, xk,6). (2.3.7) If SEP(K, & , 6 ) concludes that Xk E S ( K , a), we say that k is a feasible index and set a : = c. (2.3.8) If SEP(K, xk, 6 ) yields a vector d E R" such that (Id (1 3 1 and sup{dTx x E K } s dTxk+ 8, we call k an infeasible index and set a : = - d.

1

(2.3.9) bk:= AkaI d a T A k a , (2.3.10) x : : = x ~

+-n +1 1

bkt

Above, the sign means that the left-hand side is obtained by rounding the binary expansion of the right-hand side after p places behind the point. Since by construction x o E K , the set of feasible indices is nonempty; moreover, we can show the following theorem, cf. (41. (2.4) Theorem. Let j be a feasible index for which

< N, k feasible}.

C'X,

= max{cTxk1 0 s k

C'X,

3 S U ~ { C ~ Xx E K} - E .

Then

1

0

Cleai.j, the number N of iterations of the ellipsoid met..od is polynomial in the size of the input. One can also show that the entries of the intermediate vectors Xk and matrices A k ,0 S k S N, are polynomially bounded. Furthermore, the number 6 used to run the separation subroutine is polynomial in the input length. Thus, the ellipsoid method is a polynomial algorithm for the optimization problem for K if and only if the subroutine SEP is a polynomial algorithm for the separation problem for K. This implies, in particular, that whenever there is a polynomial separation algorithm for a class of convex bodies X there is also a polynomial optimization algorithm for .X (via the ellipsoid method). It is of particular importance that this implication also holds the other way round, namely:


331

(2.5) Theorem. Let X be a class of convex bodies. There is a polynomial algorithm to solve the separation problem for the members of YC, if and only if there is a polynomial algorithm to solve the optimization problem for the members of 3%. 0 Note that according to our definition neither the optimization nor the separation problem are solved exactly; in both cases we allow for a small error. This is necessary because the problem classes that are covered by Theorem (2.5) may also contain instances with a unique optimal solution which has irrational coefficients. But irrational numbers cannot be represented exactly. In case our class of convex bodies X is a class of polytopes, then the optimization problem for X is nothing but a linear programming problem. If in addition all members of X have a rational defining inequality system, then both the separation and the optimization problem can be solved precisely, we shall say in the strong sense. Moreover, it is also possible to construct a dual optimal solution in polynomial time. If P C R" is a polytope with rational vertices, define T ( P )to be the maximum of the absolute values of numerators and denominators occurring in the entries of vertices of P. The pair (P, T ) is called a rational polytope if T ( P )< T.The input size of a rational polytope is at least n [log T I . It is not difficult to prove that if (P, T ) is a rational polytope, then P C S(0, n T ) and if P is fully dimensional then S(ao,(nT)-2"3)C P for some point a".

+

(2.6) Theorem. Let ?€be a class of fully dimensional rational polytopes such that the optimization (or equivalently the separation) problem for YC can be solved in polynomial time. Then the following holds : (a) There is a polynomial optimization algorithm for YC in the strong sense, i.e., which for every member P E YC and every rational vector c finds a vector y E P such that cTy = max{cTx x E P } . (b) There is a polynomial separation algorithm for Yl in the strong sense, i.e., which for every member P E X,P C R", and every rational vector y either asserts that y E P or finds a rational vector c with Ilc I( 3 1 such that c T x < c T y for all x E P. In case y E P the algorithm also yields vertices xu, xI, . . . ,x, of P and rational numbers A,,, A l , . , . ,A, 3 0 such that A, = 1 and A,x, = y. (c) There exists a polynomial algorithm which for every P EX, P C R " and c E Z"provides facets aTx G b, ( i = 1,. . . , n ) of P and rational numbers A, 2 0 ( i = 1,. . . ,n ) such that A,a, = c and A h , = max{cTx x E P } . 0

I

c:=,,

c:=,

c:'=,

x:=,, 1

Case (c) of Theorem 2.6 will play an important role in the sequel, since it will provide us with a method to construct a minimum weighted clique cover from a maximum weighted stable set. A further class of convex bodies will be of interest for our purposes. Let R': be

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332

the nonnegative orthant and K C R" be a convex body such that there are reals r and R, O < r S R, witl-

Rw:flS(O,r)CKcR:nS(O. R ) , 0 sx

(2.7)

+ x E K.

y EK

(2.8)

The anti-blocker A ( K ) of K is defined by

(2.9)

A(K):={yER':(yTxS1 foreveryxEK).

It is easy to see that A ( A ( K ) )= K for a K satisfying (2.7) and (2.8),and that in case K is a polytope with vertices xI,x2,. . . ,xk then A ( K ) = { y E R: y T x i s I , i = 1.. . . , k } . Moreover, if X is a class of convex bodies satisfying (2.7) and (2.8) we set A ( x )= {A ( K ) K E XI.

1

1

(2.10) Theorem. Let .X be a class of convex bodies satisfying (2.7) and (2.8).

Then the optimization problem for X can be solved in polynomial time i f and only if the optimization problem for A ( X ) can be solved in polynomial time. 0

3. The fractional stable set problem To be able to utilize the ellipsoid method for combinatorial optimization problems one has to associate a class of convex bodies with the problem class under consideration. Natural candidates are usually the convex hulls of the incidence vectors of feasible solutions. In case of the stable set problem this is done as follows. Let G = ( V , E ) be a graph with n vertices. For every W V(G) denote by x the (node-) incidence vector of W, i.e. x ,"= 1 if u E W and x,"=O if LIEW. Then

I

P ( G ) : = c o n v { x W E R " W C V(G)isastablesetof G }

(3.1)

is called t h e stable ser polyrope of G. Clearly, every weighted stable set problem on G can be solved as a linear programming problem over P ( G ) . The polytope P ( G ) is fully dimensional, has O/l-vertices, and is contained in the unit hypercube, thus P ( G ) is a rational polytope. If we were able to design a polynomial separation algorithm for P ( G ) , then by Theorem (2.5) the weighted stable set problem would be solvable in polynomial time. Since this problem is NP-complete, we cannot expect to find a polynomial separation algorithm for P ( G ) in general. A usual approach to solve difficult optimization problems is to consider tight relaxations of the problem in question which are polynomially solvable, and then proceed by branch-and-bound methods. A natural relaxation of the stable set


333

problem is the so-called fractional stable set problem. By definition, no two vertices of a stable set are adjacent. Thus, given any clique C of a graph G, at most one vertex of a stable set can belong to C. This implies that for every clique C C V(G) and every incidence vector x w of a stable set W G V ( G ) the so-called clique inequality

is satisfied. For any graph G with n vertices we call

1

P * ( G ) : = (x E R" x, a 0 for all u E V(G) and

the fractional stable set polytope of G. P*(G) is clearly a rational polytope. Since obviouly P ( G ) C P*(G), the LP-solution over P*(G) provides an upper bound for the weight of the optimal stable set in G. For a given graph G and an objective function w : V+Z+ let us define the following parameters: a,(G):=max{wTx I x EP(G)}, a*,(G):=max{wTxI x EP*(G)}, a*(G):=max

{

x, I x E P * ( G ) } . "E"

The number a *(G) is called the fractional stability number of G, a *,(G)is called the fractional weighted stability number of G, and as mentioned earlier a, (G) is called the weighted stability number of G. By definition we have a ( G ) 6 a *(G) and a, (G) s a *,(G). At first sight the polytope P*(G)looks rather innocent. It is easy to see that its facets are the trivial inequalities xu S O for all u E V(G) and the clique inequalities & e C ~ v 6 1 for all maximal cliques C V(G) (maximal with respect to set inclusion). However, it is not known how to find all maximal cliques efficiently, even worse, there are classes of graphs (even perfect ones) such that the number of maximal cliques grows exponentially in I V(G)I. So there is no way to represent the constraint system of P * ( G ) efficiently. By Theorem (2.5) this is not necessarily crucial, since it is not the number of inequalities which matters; what matters is whether one can find a violated hyperplane in polynomial time. Since the constraint system of P * ( G )looks quite simple one might hope to find a polynomial time separation algorithm for P*(G).But this is very unlikely as the complexity of the separation problem for P*(G) is closely related to the complexity of the weighted clique problem. More precisely:

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(3.3) Proposition. Let % be a class of graphs. Then there is a polynomial algorithm to solve the weighted fractional stable set problem for every member of % if and only if there is a polynomial algorithm to solve the weighted clique problem for every member of %.

Proof. For every member G of %, the weighted clique problem can be solved in polynomial time if and only if the linear program max w'x, x E Q(G) can be solved in polynomial time, where Q(G):=conv{xC E R " ( C C V ( G ) is a clique}. By definition, the anti-blocker of O(G) is A (Q(G)) = { y E R: yTx' s 1 for all cliques C C V ( G ) }i.e. , A ( Q ( G ) )equals P * ( G ) .Thus by Theorem (2.10) the linear program max wTx, x E Q ( G )can be solved in polynomial time for every G E % if and only if the linear program max wTx, x E P * ( G ) can be polynomially solved for every G E %. 0

I

It follows from the examples in [3] that there are various classes of graphs for which the weighted clique, and hence the fractional stable set problem, are solvable in polynomial time. However, since the weighted clique problem is NP-complete for the class of all graphs, Proposition (3.3) implies that the weighted fractional stable set problem is NP-equivalent. Proposition (3.3) therefore states that considering the fractional stable set problem instead of the stable set problem does not offer considerable advantages. Moreover, the problems of computing a, (G) and a :(G) seem to be unrelated with respect to difficulty. For planar graphs ow(G) (the weighted clique number) and hence a*,(G)can be computed easily in polynomial time, while the determination of a, ( G ) for planar (even cubic planar) graphs is NP-complete; cf. [3]. So for the complementary graphs of planar graphs the determination of o, ( G ) and hence a Z(G) is NP-equivalent, while a, ( G ) can be computed in polynomial time. Although for general graphs the fractional stable set problem does not seem to be useful for computing a w ( G ) ,the situation for perfect graphs is quite particular. Namely, Fulkerson has shown the following (see also [ 11): (3.4) Theorem. Let G be a graph. Then P ( G )= P*(G)holds i f and only if G is perfect. 0

In other words, Theorem (3.4) implies that for every perfect graph G and every objective function w , a, ( G ) = a * , ( G )holds. Therefore a computationally efficient procedure determining a Z(G) would yield the desired weighted stability number. As we shall see later a , ( G ) and a*,(G) can be computed in polynomial time for perfect graphs, however, we do not make direct use of P ( G ) resp. P*(G),but rather obtain this result via a detour which will be described in the next section.

Polynomial algorithmsfor perfect graphs

335

4. The Shannon capacity, 6 ( G ) and 6,(G)

The stable set problem has found some nontrivial applications in coding theory, in particular in finding the zero error capacity of a discrete memoryless channel; cf. [12]. Let us denote by G * H the Cartesian product of the graphs G and H, i.e. V ( G .I€)= V ( G ) x V ( H ) and two vertices (u, u ) , ( u ' , u ' ) E V ( G . H ) are adjacent if and only if u is adjacent to u ' in G and u is adjacent to u' in H. G denotes the Cartesian product of k copies of G. As an interpretation, consider a graph G whose vertices are letters in an alphabet and in which two vertices are adjacent if and only if they are 'confoundable'. Then the maximum number of one-letter messages which can be sent without danger of confusion is clearly a(G),moreover, a ( G k )is the maximum number of k-letter messages such that any two of them are inconfoundable in at least one coordinate place. It is easy to see that there are at least C Y ( Ginconfoundable )~ k-letter words, but in general there may be many more such words. To measure the largest rate at which one can transmit information with an error probability exactly equal to zero, Shannon [ 121 introduced the following number:

which is now called the Shannon capacity of graph G. From the fact that a ( G k " ) 3a ( G k ) a ( G e )it directly follows that

O(G) = lim k Va(Gk)

(4.2)

-m

and, since ( Y ( G )s~ a ( G k ) ,that

Shannon f12] obtained an upper bound for O(G) by showing

O(G)Sa*(G).

(4.4)

However, both inequalities a ( G ) 0; moreover, since B' is integral, det(B') 3 1. It follows that

2 A:

i=l

= tr(8') = na(G) and

fi A:

= det(B')a 1

i=l

Using a rough estimate we obtain A L 2 (fly:,: A :)-I vations imply that the matrix

2 ( n a (G))-"+'. These

obser-

is a positive definite matrix contained in B ( G )whose smallest eigenvalue A { is not smaller than (na(G))-"and for which b: = a ( G ) .Hence, if we subtract (na(G))-"from every element of the main diagonal of B" and then multiply the - n ) we obtain a matrix B which has resulting matrix by (na(G))"/((na(G))" trace one, is positive semidefinite, and satisfies b,)= 0 if i,j E E ( G ) . Thus B E SB(G),and since a ( G ) 2 2

c:l=l

and it follows from (4.13) that for a critically imperfect graph G (4.16)

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Polynomial algorithmsfor perfect graphs

We now prove that 6 ( G ) < a * ( G ) . Padberg [ll] has shown that for a critically imperfect graph G, a * ( G )= cu(G)+l / w ( G ) ,and that there is a unique point y E P * ( G ) ,namely,

c:=,

which satisfies y , = a * ( G ) .B y (4.6)there exists an orthonormal representation ol,. . . ,u, of G and a vector d of unit length such that 6 ( G )= (dTu,)2. As shown in the section following formula (4.6), the vector x E R " with x, = (dTo,)', i = 1 , . . . ,n is contained in the fractional stable set polyhedron P*(G).Now suppose that 6 ( G )= a * ( G ) ,then

c:=,

hence by the uniqueness of y we have , 1 -- - yi = xi = (d'u,P, i = 1,. . . ,n. 4G) Thus for the orthogonal representation u,, . . . , ZI, of G and the vector d E U we have

and therefore formula (4.5) implies that 6(C?)Sw ( G )= a ( G ) . However, the complementary graph of a critically imperfect graph is critically imperfect too, so 6 ( G ) s a ( G ) contradicts (4.16). This implies that 6 ( G ) cannot equal a * ( G ) . Summing up we have shown that for any critically imperfect graph G

1

a ( G )< 6 ( G )< LY *( G ) = ( G )+ w(G)

(4.17)

a

Every imperfect graph contains an induced subgraph which is critically imperfect, i.e., a subgraph for which (4.17) holds, while for every induced is satisfied. This subgraph G' of a perfect graph G, a(G')= 6(G') = a *(G') implies the following characterization of perfect graphs. (4.18) Theorem. A graph G is perfect if and only if the following holds:

a ( G [ W ] ) = 6 ( G [ W ] forall ) WcV(G). 0 Our discussion above yields a further characterization, namely, a graph G is perfect if and only if a * ( G W [ ] )= 6 ( G [ W ] )for all W C V ( G ) . The number 6 ( G ) is not necessarily equal to a ( G ) , even worse, Konjagin (unpublished) has constructed a sequence of graphs G, with n vertices such that

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340

a ( G n )= 2 and 8 ( G m ) - + mThis . implies that there is no function f at all such that

6 ( G ) S f(cr(G)) holds for all graphs G. It seems to be an interesting problem if there exists a polynomially computable function q ( G ) and a function f such that a ( G )S d G ) f(Q(G)). An efficient way to calculate 8 ( G )provides us only with a good algorithm for the unweighted stable set problem in perfect graphs. In order to cover the weighted case too we now generalize 8 ( G )t o a weighted version 8,(G). Assume that a graph G = (V, E ) and a weight function w : V - + Z, are given. Define the graph G, to be the graph arising from G by replacing each vertex u of G by w. pairwise nonadjacent new vertices and where two vertices of G, are adjacent if and only if their originals in G are different adjacent vertices. This construction implies that a, (G) = (Y (G, ) holds. Moreover, Lovdsz [9] has shown that if G is perfect, then G, is also perfect (in fact, this is the key lemma for the perfect graph theorem). Hence for perfect graphs we have a , ( G ) = 8(G,) = a * , ( G ) Therefore . we define

O , ( G ) : = S(G,).

(4.19)

Note, however, that the existence of a polynomial algorithm for calculating 9 ( G ) does not give a polynomial algorithm for 6,(G) by applying this algorithm to G,, since no algorithm making up G, from G and w is polynomial in the input length O (I E I + log 11 w ]Im). A characterization of 8,( G )using the set SB(G)defined in (4.12) and avoiding this construction is given in the following theorem. (4.20) Theorem. Let G = ( V, E ) be a graph and w : V -+ B , a weight funcfion,

then (4.21)

Proof. Let M be the maximum in (4.21). We first prove M 6 6,(G) using formula (4.13) for 8(G). Suppose B = ( 6 , ) E 9 ( G ) . Replace every entry bij by a ( w i , wj)-matrix all of whose entries are (wiwj)-i'2bij to obtain an (r, r)-matrix B', where r = wi. B' is clearly symmetric and satisfies

c:=,

tr(B') =

2 bii =

,=I

i=l

bii = 1;

moreover, the definition implies that bh = 0 if i,jE E ( G ) . It is also easy to see that B' is positive semidefinite. Furthermore, simple calculation shows " bii = G j b i j

2

#.I = 1

i.j = I

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Polynomral algorithmsfor perfecr graphs

which implies M G 6,(G). Conversely, with each matrix B ' E B(G,) we can associate a matrix B E %(G) such that b:, s G J 6 , holds, which proves 6, (G)<M. Namely, take B' E B(G,). Replace the (w,, w,)-submatrix induced by the copies of i and j, by the sum of its entries divided by V F , , and eventually add a nonnegative number to any diagonal entry to make the trace equal to one.

c:,,=, c:,=,

5. A separation algorithm for a class of positive semidefinite matrices with trace one

In this section we shall describe a polynomial time separation algorithm for the class of positive semidefinite matrices %(G) defined in (4.12). Every set 9 ( G ) is clearly convex and bounded, but not fully dimensional. Since the ellipsoid method, as described in Section 2, can only be applied to convex bodies, we have to replace the sets B(G) for technical reasons by fully dimensional ones. For every (n, a)-matrix B = (b,,),i.e., B E R""", and every graph G we define n + (z) - I E ( G ) I - 1. the following projection operation R"""+ R" where ii:= Discard from B all elements below the main diagonal; all lE(G)I elements b,,, i < j, corresponding to different adjacent vertices i,j E V ( G ) ;and the element b,,,,. Denote the vector of R" obtained from B in this way by B, i.e., B is an ii-vector whose components are indexed by F = {ii i = 1,. . . , n - 1) U {zj i < j and ij$Z E ( G ) } . Conversely, for every vector B E R E we define an extension operation n-1 R" + R n x n by setting b,,:=6,, i = 1,. . , , y1 - 1; b,.:= 1 b,, ; b,, = bJZ= 0 for i , j E E ( G ) ; b,, = b,, = 6,for i , j E E ( G ) , i < j. By definition the (n, n)-matrix B obtained by extending B E RE is a symmetric matrix with trace one. Now set

1

1

c,=,

I

%I(G):={BE R x B E B(G)};

(5.1)

a ( G ) is convex, since it is the projection of a convex set. We now prove that g ( G ) is fully dimensional and bounded, i.e. that a ( G ) is a convex body. Denote by. B, the projection of the matrix 1 B.:=;

I,,E B(G)

(Zn is the (n,n)-identity matrix); then the following holds:

Set r = (nZ 0 for 9, ( G ) are given, then the following algorithm THETA ( G ,w, E, 7 ) finds a number T with 17 - aW (G)(< E. (6.3) Algorithm. THETA(G, w, E, 7).The graph G = ( V ( G ) ,E ( G ) ) , 1 V ( G ) )= n 3 2, the natural numbers w,,i E V ( G ) ,and the rational number F > 0 are the input of the algorithm, while the number 7 is the output of the algorithm.

vGj,

(6.3.1) Approximate the numbers 1 i s j s n, by rationals u,j satisfying (6.2) and whose denominators are at most 2 n ( n + 1 ) / ~ Set . ri,, := u,, - u,,,

i

=

1 , . .., n

-

1 and ri,j: = 2u,, for i

< j, i, j E E ( G ) .

(We now approximate the optimum value of the program max{GTB B E B ( G ) }+ unn up to an error ~ / using 2 the ellipsoid method.)

1

(6.3.2) Set r = l / n z f i , R = 1, E : = ~ / 2 -~ / 2 n ( + n 1) and define the paramters N, 6, p as in the ellipsoid method (2.3).(For the choice of the radii, cf. (5.2), the accuracy E is chosen according to the previous discussion.) (6.3.3) Set Ao:=RZZ,and choose as center xo of the first ellipsoid the projection B, E B ( G )of (l /n )Z n .

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(Recall that ii = n + (;) - [ E ( G ) (- 1, and that by (5.2) B, is an interior point of $%(G).) (6.3.4) for k = 0 to N - 1 do; 1. Run the separation algorithm SEP(G,xk) defined in (5.5). 2. If x k E & ( G ) then set a:= a. 3. If X k E B ( G ) and if 0 is the vector returned by SEP(G,xk), cf. (5.5.6), then set a:= - D. 4. Make the updates of the ellipsoid method as described in (2.3.9)-(2.3.11). End; (6.3.5) Let cr be the value of the best feasible solution of rnax{z'i'B B E a ( G ) } found in (6.3.4). Set T : = m + u,, and return T. 0

I

Algorithm (6.3) describes how we can approximte 9, (G) in polynomial time. Letting w be the vector all of whose components are one, we can use algorithm THETA to compute 6 ( G ) up to any given accuracy in polynomial time for every graph G. So 9( G) is not only well-characterized by the formulas (4.3,(4.6), (4.9) and (4.13), it is also well-behaved computationally. Theorem (3.2) and Theorem (4.20) imply that for perfect graphs the numbers a,(G) and & ( G ) coincide. Moreover, since our weight function w is integer valued, we know that the value a, (G) of the optimum weighted stable set is an integer. Therefore, in order to find the optimum value of a weighted stable set problem on a perfect graph we only need to approximate 6,(G) up to an error E c f with the algorithm THETA(G, w, E , T ) and round T to the next integer to obtain a, (G). We can also use the algorithm THETA to find a maximum weighted stable set explicitly. This goes as follows. Let a graph G = (V, E) with n z= 2 vertices, weights wi E Z, for all i E V and an accuracy O < E s $be given. (6.4) Algorithm. STABLESET(G, w, E ) .

(6.4.1) Initialization and first guess for a(G): 1. Run THETA(G, w, E , T ) and round T to the next integer, say f. (Clearly, r 3 a ( G ) and if G is perfect then t = a (G).) 2. If 1 r - T 13 E, then stop and conclude that G is not perfect. 3. Call all vertices of G unlabeled. (6.4.2) Termination check If all vertices of the present graph G are labeled, then do:

349


1. If V ( G )is not stable then stop and conclude that G is not perfect. 2. If V ( G )is stable then V(G) constitutes a maximum weighted stable set of the original graph. stop! (6.4.3) Choose an unlabeled vertex z, E V and tentatively remove u from G, i.e., set G‘ = G - u and set w L = w, for all u E V \{ u } . (6.4.4) Run THETA(G’, w ’ , E, T ) and round T to the next integer, say s. If I s - T 12 E , then stop and conclude that G is not perfect. (6.4.5) If s = t, then remove u definitely, i.e., set G = G ’ and w

= w’.

(6.4.6) If I V(G)I = 1, then label the remaining node. (6.4.7) If s < t, then label u. (6.4.8) Go to (6.4.2). 0 To prove the correctness of the algorithm suppose first that G is a perfect graph. Then for every induced subgraph G’ of G, 6,.(G’) = a,.(G’). In step (6.4.1) we approximate & ( G ) by F G $ , and, therefore, rounding 7 to the next integer gives the true value t for a ( G ) .Now we remove a vertex u from G. If the number s calculated in (6.4.4) satisfies s = f then we know that G - u contains a stable set which is a maximum weighted stable set of G, so we can remove u without distroying all optjmum solutions of the stable set problem for G, and we can continue with this procedure. If however s # t, then all optimum stable sets of G necessarily contain vertex u. Thus we label u, keep u in our vertex set and continue. This way we will finally end up with a graph G ’ whose set of vertices is labeled, i.e., none of the vertices can be removed without reducing s = a,.(G‘)= a, ( G ) = c. This means that every vertex of G’ is contained in all optimum stable sets of G’. In other words, the vertex set of G ’ is itself a stable set, and since a,.(G’) = a , (G), this vertex set is a maximum weighted stable set of G. Thus if G is perfect, then STABLESET will produce an optimum weighted stable set of G. If however G is not perfect, then STABLESET may detect the imperfectness of G but may also deliver a maximum weighted stable set (without recognizing the imperfectness of G). If in step (6.4.1) or (6.4.4) we find that ] t - T I 3 E resp. I s - T 1 2 E then the interval ( T - E , T + E ) contains no integer. Since cu,.(G’) is an integer for all induced subgraphs G‘ of G and THETA guarantees 6,.(G‘)E (T - E , T + E ) this implies a,.(G’)# IL(G‘),i.e., by Theorem (4.18) we can conclude that G is not perfect. It may however happen that in every step (6.4.1) and (6.4.4) the approximation 7 of 6,JG’) is in the &-neighborhood of an integer and we will end up in step (6.4.2) with an induced subgraph G’ of G whose vertex set V’ is labeled. If V‘ is not a stable set, then V’ is not a solution of our

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stable set problem. Since the algorithm works for perfect graphs, we can conclude that G is not perfect. If V' is a stable set we have to show that V' is in fact a maximum weighted stable set of G. This can be seen as follows. Suppose G" is the last subgraph of G created during the algorithm such that a vertex, say u, was definitely removed from G". Then we know that all other vertices of G" will finally be labeled, so V(G")= V' U { u } . Moreover, u was removed because the number s obtained in (6.4.4) by running THETA(G"- u, w ' , E ,T ) satisfies s = t. Since V' is stable, G' = G"- u is a perfect graph, so we have s = a,.(G'), and since t is an upper bound for LZ, (G), s is the weighted stability number of G. This proves our claim. The following (imprecise) argument shows that in case of imperfect graphs all outcomes described above are possible. Consider the pentagon G, then 9(C,) = L 5 = 2.236.. . . Suppose we run STABLESET with E = 4, then the T we get in step (6.4.1) may equal 2.1 or 2.6. If T = 2.1 then t = 2 and the algorithm will continue finding a maximum stable set of C5. If T = 2.6 then t = 3, and since the removal of every vertex from Cc results in a perfect graph we shall get s = 2 every time we execute step (6.4.4). This means that finally all vertices of C, will be labeled, but these of course do not solve our problem. If however we had chosen E =0.1, then ~ € ( . \ / 5 - 0 . 1 , V ' ~ + O . I ) a n d w e o b t a i nt = 2 a n d l r - ~ I s 0.13 > E . This implies that the algorithm would stop in step (6.4.1) concluding that C, is not perfect. Thus algorithm STABLESET has two possible outcomes. Either a maximum weighted stable set of G is found or imperfectness of G is proved. Note that in the former case it does not prove perfectness. STABLESET can also be used to find a maximum weighted clique of a perfect graph G. We simply run STABLESET on the complementary graph G which by the perfect graph theorem is perfect again. Summarizing the observations of this section we obtain the following theorem.

(6.5) Theorem. (a) There exists an algorithm which for any graph G = ( V , E ) , 1 VI 2 2, any weight function w : V-, 8, and any rational E > 0 ,findsa number r such that I T - a,(G)J < F holds. The running time of this algorithm is polynomial in I V ( , [log((w 11-1 and rlog I1

f

11-1 .

(b) There exists an algorithm which for any perfect graph G = ( V , E ) , I V I 3 2 and any weight function w : V+H+ finds a maximum stable sef (resp. maximum weighfed clique) and the running time of which is polynomial in I VI and [log It w 11-1 '

Algorithm THETA and inequality (4.16) can be combined to design a polynomial time nondeterministic algorithm which checks the imperfectness of a


35 1

given graph. Namely, suppose G' is a critically imperfect graph with n vertices, then choosing a suitable E , e.g., E = in-'", we run THETA(G', e, E , T ) where e is the vector all of whose components are one. By Theorem (2.4), the choice of F , and inequality (4.16) the number T we obtain satisfies T E(8(G') - E

, ~ ( G ' E) )+,

T

a ( G ' ) , T + E < cu(G)+ 1. Since lo g ( ( €(Ix is polynomial in n, THETA runs in time polynomial in n. In other words, given a critically imperfect graph, we can verify its imperfectness in polynomial time. As every imperfect graph contains a critically imperfect graph, say G', we can guess this graph G ' and then apply the algorithm described above. This shows that verification of imperfectness is an NP-problem, hence verification of perfectness is a co-NP problem. Note that if the strong perfect graph conjecture is true, then this fact is trivial, since checking imperfectness would then be possible by guessing an odd hole or antihole.

7. A polynomial algorithm for the weighted clique cover and coloring problem for perfect graphs

The separation algorithm for g ( G ) presented in Section 5 provides us - as shown in Section 6 - via the ellipsoid method with a polynomial time algorithm for solving the weighted stable set problem for perfect graphs. Seen from a different point of view this means that the class of linear programming problems maxcTx, x E P ( G ) = conv{x W stable set in G}, G a perfect graph, is solvable in polynomial time. Since P ( G )is a fully dimensional rational polytope, Theorem (2.6) implies that the optimization problem as well as the separation problem for P ( G ) ,G perfect, are solvable in polynomial time, even in the strong sense. By Theorem (3.2), for a perfect graph G the stable set polytope P ( G ) equals the fractional stable set polytope P * ( G ) , thus for this class of graphs we can decide in polynomial time whether a given vector y belongs to

1

P ( G ) = P * ( c ) = ( ~ ( ~all, ~ o E ~v ~, ~

c x, s I

"EC

for all cliques

c c v(G)/ .

A different approach to solving the separation problem for P ( G ) not using

Theorems (2.5) and (2.6) is of course to apply the algorithm which finds a maximum weighted clique, where the given vector y 3 0 is used as the vector defining the objective function. If the maximum clique, say C, satisfies y r ~ ='

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cuEcyu d 1 then y E P ( G ) ,otherwise this clique inequality provides a separating hyperplane. Since the optimum clique algorithm is nothing but the optimum stable set algorithm applied to the complementary graph G (which is also perfect), we can use the algorithm STABLESET directly to solve the separation problem for P ( G ) , G perfect. Theorem (2.6) has a further important consequence. Since the optimization problem for the class of rational polytopes P ( G ) , G perfect, is solvable in polynomial time we can find facets of P ( G ) and rationals A, 2 0 satisfying the conditions of statement (c) of (2.6). Since the facets of P ( G ) are of the form - x u SO, u E V ( G ) ,and cvGcx, G 1, C c V ( G )a maximal clique, Theorem (2.6) (c) implies that for any objective function w : V(G)-+Z+ we can find in polynomial time (maximal) cliques C C V(G) and positive rational numbers A,, i = l , ..., r S l V ( G ) l such that

Suppose for a node u E V ( G )strict inequality holds in the above inequality, say u,:=C:=l,,,c,A, - w , > 0, then pick a clique, say C,, which contains u. If A, s u, then replace C, by the clique C, \ { u } , otherwise add the new clique C, \ { u } as clique C,&, to our list of cliques and define new parameters as follows: A,:= A, - u L , A,,, := u,. Then the sum of the A,'s still equals a, ( G ) and the gap u, of the inequality corresponding to u is either zero or is strictly reduced. By continuing this process we end up wirh a list of cliques C,,, . . , C, and positive rationals A , , . . . A, such that

.

2 A, = a,(G). I

=I

8

=I.

A, = w , for all u E V(G). C i E C ,

Note that in the algorithm described above only those vertices u E V ( G )were considered for which the inequality - x, d 0 had a positive multiplier A,. Since for every such vertex at most one additional clique was added, we still have r 6 1 V(G)\. By definition, for a perfect graph G the stability number a ( G ) equals the clique cover number p ( G ) . Moreover. since for a perfect graph G the graph G, (cf. Section 3) is also perfect and as a,(G)= a(G,), p w ( G ) =p ( G , ) , the weighted clique cover number pw ( G ) equals the weighted stability number a w ( G ) .This implies that for a perfect graph G, algorithm THETA (or STABLESET) also calculates pw (G), thus, by definition, there exist integers A, > 0 which satisfy (7.1). Note that the numbers constructed by the algorithm of Theorem (2.6) ( c ) (plus scaling afterwards) need not be integral in general.


353

However, we can find such integers for perfect graphs. We first show how this can be done in the cardinality case, i.e., w = (1,. . . ,l)T. Assume that G = ( V ,E ) , I V 1 3 2, is a perfect graph and we want to find a clique cover of G. (7.2) Algorithm. Cardinality clique cover.

(7.2.1) Apply the algorithm of Theorem (2.6) (c) to find cliques CiC V and (possibly nonintegral) rationals hi > 0, i = 1,. ..,t satisfying (7.1). (We claim that every clique C, with hi > 0 intersects every stable set of G of cardinality a ( G ) . Suppose C, is such a clique and W C V is a maximum stable set with W f l Ci= 0. Then (7.1) implies

i#/

a contradiction. Therefore we continue as follows.) (7.2.2) Remove clique CI from G, i.e., set G:= G - CI. If V ( G )= 0. stop. Otherwise go to (7.2.1). (Note that the graph G' obtained from G by removing clique CI satisfies a ( G ' )= a ( G )- 1 since every maximum stable set of G will lose one vertex. So after exactly a ( G )executions of (7.2.1) and (7.2.2) we have found a ( G )cliques which cover G.These cliques are those which have been removed in (7.2.2). Note also that every vertex of V is contained in exactly one such clique and that these cliques are not necessarily maximal cliques of G.) 0 Since the algorithm of Theorem (2.6) (c) can be shown to be polynomial in I V I the overall running time of algorithm (7.2) is also polynomial in I V I. We shall now extend this algorithm to the weighted case. Assume that a perfect graph G = ( V ,E ) and a weight function w : V +H, are given. (7.3) Algorithm. Weighted clique cover.

(7.3.1) Apply the algorithm of Theorem (2.6) (c) to find cliques C C V and (possibly nonintegral) rationals hi 0, i = 1 , . . . , f, satisfying (7.1).

=-

(7.3.2) Set A ::= LA,J, i = 1,. ..,t,

and construct the graph G,.. (Since hi - A : < 1 we obtain from (7.1)

M.Grotschel el al.

so the graph G,. obtained by replacing every node of v by w l nonadjacent copies and linking two nodes in G,. by an edge if their originals in G are adjacent has less than I V )z vertices. This implies that G,, can be constructed from G in time polynomial in I V ( and [log II w 11-1

(7.3.3) Apply algorithm (7.2) to G,, to obtain cliques D : , i = 1,. . . ,a(G,.) covering each vertex of G,, exactly once. (Since every clique D ;of G,. contains at most one copy of every vertex u E V ( G ) ,every DI corresponds to a clique, say D,, of G. Note that for 0: # 0; the corresponding cliques D , , 0,of G may be identical.) (7.3.4) Construct the cliques DI,. . . ,Do(G,,) of G corresponding to the cliques DI,. . . ,DL(G,.) of G,.. Let B I , .. .,B, be the different cliques occurring in the sequence D,, i = 1 , . . .,a(G,.) and let p,, j = 1 , . . . ,r, be the number of times clique B,occurs in the sequence D,, i = 1,. . . ,a(G,.). Then proceed as follows: Set k = t and for j = 1to r do; If B, is equal to one of the cliques C,, i E 11,. . . ,I } , then set A: := A + p,. Otherwise set k : = k + l , AL:=p, and Ck:=B,. 0

We claim that the cliques C, and integers A:, i = 1,. . . ,s, defined in step (7.3.4) solve the clique cover problem considered. Obviously, in the above algorithm every vertex u E V is covered w, - w : times after the execution of step (7.3.2). By applying algorithm (7.2) to the graph G,. and making the construction described in (7.3.4) every vertex will be covered a further w L times. So the cliques C1,. . . ,C, and integers A :, .. . , A I satisfy

Similarly, note that G,. is designed in such a way that a ( G , . ) = a,(G)-C:=I(A,- [ A , ] ) holds. Since G is perfect G,, is also perfect, so p(G,.) = XI=, p, = a(G,.) which implies that the A: defined in (7.3.4) satisfy ,=1

A', = a , ( G ) = p, (G).

Thus algorithm (7.3) produces the desired solution of the clique cover problem for a perfect graph. Since the algorithm STABLESET, the algorithm of Theorem (2.6) (c) and the algorithm (7.2) run in time polynomial in 1 V(G)1 and [logl)w 11-1 the overall


355

running time of algorithm (7.3) is polynomial in I V(G)I and [log\(w 11-1 for every perfect graph G and every objective function w : V(G)+Z+. As before it is now easy to obtain a polynomial time algorithm for the weighted coloring problem for perfect graphs. Since the weighted chromatic number x w ( G )equals the weighted clique cover number p , ( G ) of the complementary graph G we simply apply algorithm (7.3) to the perfect graph G which will yield the desired optimum weighted coloring of G.

8. Conclusions

In the previous section we have described polynomial time algorithms for various linear programming problems on perfect graphs. All these algorithms are based on the ellipsoid method and use a polynomial time separation algorithm for a convex, nonpolyhedral set. Although these algorithms are polynomial (and thus are theoretically good) we do not recommend them for practical use. Just for curiosity we have done some computational experiments with the separation algorithm for G(G) described in Section 5. As expected, the numerical problems were such that even for small problem sizes, say V(G)I equal to 10 or 20, it was almost impossible to obtain a correct answer. An alternative approach is to use (4.9) for the design of a polynomial algorithm to compute 6(G). This amounts to minimizing a convex function on an affine space. In principle, this can be done by the ellipsoid method in polynomial time. In practice, it is probably better to use some simpler descent method. The first experiments with this dual approach seem to be more promising. Our analysis of these problems should be viewed as a theoretical contribution showing that certain programming problems for perfect graphs are indeed polynomially solvable. Future research should be directed toward finding practically good algorithms for these problems. These algorithms should have a more combinatorial nature and should not suffer from the numerical instability (due to our present-day computer technology of fixed precision arithmetic) of the ellipsoid method and the separation problem for B ( G ) .

I

References [ l ] V. Chvital, O n certain polytopes associated with graphs, J. Comb. Theory, Ser. B 18 (1075) 138-154. [2] P. Gacs and L. Lovasz, Khachian’s algorithm for linear programming, Math. Program. Studies 14 (1981) 61-68.

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131 M.R. Carey and D.S. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979). [4] M. Grotschel, L. Lovasz and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981) 169-197. [ 5 ] W. Haemers, On some problems of Lovasz concerning the Shannon capacity of a graph, IEEE Trans Inform. Theory IT-25 (1979) 231-232. [6] W.-L. Hsu, How to color claw-free perfect graphs, Ann. Discrete Math. 11 (1981) 189-197. [7] W.-L. Hsu and G.L. Nemhauser, Algorithms for minimum coverings by cliques and maximum cliques in claw-free perfect graphs, Discrete Math. 37 (1981) 181-191 (this volume, pp. 357-369). [8] L.G. Khachiyan, A polynomial algorithm in linear programming, Dokl. Akad. Nauk SSSR 244 (1979) 1093-1096 (English transl. Soviet Math. Dokl. 20 (1979) 191-194). [9] L. Lovasz, Normal hypergraphs and the perfect graph conjecture, Discrete Math. 2 (1972) 253-267 (this volume, pp. 29-42). [lo] L. Lovasz, On the Shannon capacity of a graph, IEEE Trans. Inform. Theory IT-25 (1979) 1-7. [ I I] M.W. Padberg, Almost integral polyhedra related to certain combinatorial optimization problems, Linear Algebra & Appl. 15 (1976) 6Y-88. [ 121 C . Shannon, The zero error capacity of a noisy channel, IRE Trans. Inform. Theory IT-2 (1956) 8-19. [I31 N.Z. Shor, Convergence rate of the gradient descent method with dilatation of the space, Kibernetika 2 (1970) 80-85 (English transl. Cybernetics 6 (1970) 102-108). [ I41 D.B. Yudin and A. S. Nemirovskii, Informational complexity and effective methods of solution for convex extremal problems, Ekonomika i Mat. Metody 12 (1976) 357-369 (English transl.: Matekon: Transl. of Russian and East European Math. Economics 13 (1976) 24-25).


ALGORITHMS FOR MAXIMUM WEIGHT CLIQUES, MINIMUM WEIGHTED CLIQUE COVERS AND MINIMUM COLORINGS OF CLAW-FREE PERFECT GRAPHS* Wen-Lian HSU Department of Industrial Engineering and Management Science, Northwestern Universiry. Evanston, Illinois, V.S.A.

George L. NEMHAUSER School of Operations Research and Industrial Engineering, Cornell University, Ithaca, New York, U.S.A.

For the class of claw-free perfect graphs with weights on the vertices, we give efficient algorithms for finding a maximum weight clique and a minimum weighted clique cover. We also present an algorithm for minimum cardinality coloring of these graphs, but the minimum weighted coloring problem remains unsolved.

1. Introduction

One of the unsolved problems for perfect graphs is to find efficient combinatorial algorithms for the maximum weight independent set, maximum weight clique, minimum clique cover and minimum vertex coloring problems. * * Graphs that do not contain claws, odd holes and odd anti-holes are known to be perfect (see [13], [9], [3]). We consider this family of perfect graphs. Efficient combinatorial algorithms for the maximum independent set problem on claw-free graphs, including the weighted case, have been given by Minty [ 121 and Sbihi [14]. However, one cannot expect to have efficient algorithms for the maximum clique or minimum clique cover problems on claw-free graphs because these problems are NP-hard ([7]). The edge coloring problem on general graphs can be reduced to vertex coloring on claw-free graphs [7]. Holyer [15] has shown that minimum edge * This research has been supported, in part, by National Science Foundation Grant ECS-8005350 to Cornell University. The results of this paper have appeared in Hsu’s unpublished Ph.D. Dissertation 17) and also in Refs. [6], [8]. * * Grotschel, Lovasz and Schrijver [4] use Khachian’s ellipsoid method [ l l ] to obtain polynomial algorithms for a number of combinatorial optimization problems including the four problems just noted on perfect graphs. Although these results are theoretically very significant, the ellipsoid algorithm is not practically efficient. 351

358

W.-L.Hsy G.L.Nemhauscr

coloring of general graphs is NP-hard and consequentially minimum vertex coloring on claw-free graphs is NP-hard as well. However, when we assume perfection as well as claw-freeness, these problems become tractable, as we have shown in our recent papers ([6], [8]). Here we present the combinatorial algorithms of these papers for maximum weight cliques, minimum weighted clique covers and minimum cardinality vertex-colorings of claw-free perfect graphs. The coloring problem in the weighted case remains unsolved.

2. An algorithm for maximum weight cliques on claw-free perfect graphs The algorithm is a simple consequence of the following lemma.

Lemma 1. Let G be a claw-free perfect graph and let u be any vertex of G. Let N ( u ) be the neighbors of u and let Gnbe the subgraph of G induced on N ( u ) . the complement of G O , is bipartite.

a",

Proof. Since the subgraph induced by N(u) U { u } is claw-free and every vertex of G" is adjacent to u, Gncannot have triangles. Furthermore, Gnhas no odd anti-holes so that Gohas no odd holes. A graph without triangles or odd holes has no odd cycles. 0

Since G" is bipartite, we can find a maximum weight vertex packing (independent set) P, on Go efficiently by the O(n3)maximum flow algorithm of Karzanov [lo]. f, U { u } is a maximum weight clique in G that contains u. Thus by enumerating over all u E G we can find a maximum weight clique of G in O ( n 4 )time.

3. An algorithm for minimum weighted clique covers Let A be the clique-vertex incidence matrix (abbreviated as clique matrix) of the perfect graph G, i.e., each column of A corresponds to a vertex of G and each row of A corresponds to a clique of G. Chvital [l] proved that every extreme point of the polyhedron P = {x E R : Ax d 1) is a (0,1)-vector, from which it easily follows that there is a one-to-one correspondence between extreme points of P and packings of G. Hence we can formulate the maximum weight vertex packing problem on a perfect graph as the linear program*

1

' Rows of A that correspond to non-maximal cliques are obviously superfluous. However, we retain them in the problem description because our clique cover algorithm uses non-maximal cliques.

Algorithms

(WW

359

max w . x, Ax s 1, x 30,

where w is the vector whose components are the vertex weights wt(u) for all u E V. Without loss of generality, we will assume that w is strictly positive since vertices with non-positive weights can always be assigned a value of zero in some optimal solution. The dual of the above program is the linear programming formulation of the weighted clique cover problem WCC)

min 1 . y, yA 2 w, y 30.

When w is an integer vector, perfection implies that (WCC) always has an integral optimal solution (see [2]). However, our approach can solve (WCC) for any real w 3 0. Of course, if an integral y is required then w is also required to be an integer vector. Since G is claw-free, Minty’s algorithm can be used to find a maximum weight packing P of G. Our approach is to begin with P and then to find a feasible solution y * to (WWC) with the property that 1 . y * = wt(P), which implies (by weak duality) that y * is an optimal solution to the weighted clique cover problem.* We use the fact, also implied by duality, that for u E P there exists a clique C containing u and an E , 0 < E < wt(u), such that P is a maximum weight packing on graph G with weights given by wt(u)- E for u E C and wt(u) otherwise. The difficult problem is to find such a weighted clique efficiently. A recursive application of the weighted clique-finding procedure on reduced graphs will yield an optimal weighted clique cover for graph G. To make this idea precise, we first define graph reduction and then the construction of the clique cover for G from the weighted cliques. Finally, we will give the algorithm for finding the weighted cliques. ) V ( H 1 and ) E ( H 2 )C E ( H I ) .Let Let HI,Hz be two graphs such that V ( H 2 C wtl( .) and wtz(. ) be weight functions on the vertices of H1 and H z respectively. Define the reduced graph of H I by H 2 , HIIH2,to be the graph H’ with

I E ( H ’ )= { e 1 e = (x, y ) E E ( H ~ )x,, y E v(H’)I,

V ( H ’ )= { u u E V ( H I )and wtl(u) > wt2(u) if u E V ( H 2 ) } ,

* We also used this approach in the unweighted case [9]; however, the technique used here to construct the clique cover is quite different from the technique that we gave in [9].

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360

and weight function wt'( - ) defined on V ( H ' ) by wt'(u)

wt,(u)

if u E V (HI ) \V ( H 2 ) ,

wtl(u)-wt2(u)

if u E V(H,)n V ( H 2 ) .

=

Given a maximum weight packing P and u f P, the algorithm finds a clique C:, with weight yc: > 0 such that p is a maximum weight packing on the reduced graph G I C f , .The algorithm continues the clique finding and reduction procedure until the weight of u is reduced to zero. At each step the optimality of P is maintained. Let C, be the set {C:}rA;'of weighted cliques containing u that have been deleted during the process and call the reduced graph at this stage G,. By construction

and P\{w} is a maximum weight packing in G,. Then we replace G by G, and P by P \ { u } and apply the algorithm recursively. When P becomes empty, the reduced graph will also be empty. Hence a recursive application of the algorithm will give rise to I P 1 sets { C",},u, E P, of weighted cliques such that each vertex u of G is covered by at least wt(o) cliques and for each u, E P K(u

c'

yr;,= wt(uJ) in G.

,=I

Thus

so that the set u,",{Cu,} is an optimal solution to the weighted clique cover problem. During the procedure, we also have to consider a graph with weights on the edges, which is defined as follows. Let G be a weighted claw-free perfect graph with positive weight on each vertex. Let u be a vertex in a maximum weight packing P in G. Since G is claw-free, 10 n N ( u ) l S 2 for any packing Q in G. From Lemma 1, we know that Go is bipartite. For all (x, y) E E ( G " ) define edge weights by wt((x,y))=max{wt(Q)( Q is a packing in G, { x , y } E 0)-wt(P\{u}) and for all u E N ( u ) , define tower vertex weights by

1

wt(u) = max{wt(Q) Q is a packing in G, Q -wt(P\{u}).

n N ( u )= { u } }

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361

Note that wt((x,y)) and wt(u) can be found by Minty's algorithm [4]. Construct a weighted graph H, as the graph Gowith usual vertex weights as in G and edge weights and lower vertex weights as above. It is easy to show that P is optimal if and only if (1) wt(u) s wt(u) for all u E N ( u ) ; (2) wt((x, y ) ) s wt(u) for all (x, y ) E E ( H , ) . Therefore to maintain the optimality of P for the reduced graphs, we select the cliques {Cl}and their weights {yc;,}so that in the reduced graphs (1) and (2) are satisfied. The following algorithm selects the collection of cliques C,,: begin let G" be the subgraph of G induced on N ( u ) ; construct H, from G ;

C"+0; while wt(u)>O do begin w , +maximum lower vertex weight in H , ; w z +maximum edge weight in H, ; Wmax +max{ w 1, wzj; if w,,, < wt(u) then comment this is case 1 in the proof; begin let C be the clique { u } ; yc +wt(u)- w,,,; Cu + Cu U { C } ;wt(u)+ Wmax; end else comment now w,,, = wt(u); this is case 2 in the proof; begin A , +{v u E V(H.) and wt(v) = wmax}; comment A , is the set of vertices that must be included in the next clique; A r + { x x E V ( H , )and 3 y E V ( H , ) s.t. wt((x, y)) = w,J; E~ {(x, y wt((x, Y 1) = W m a J ; comment our next clique must include one end point of each edge in E Z ; HI+ a graph with V ( H I ) =A , U A 2 and E ( H , ) = E z with the same vertex weights and edge weights as in H , if A , = 0 then begin comment now w,,, = w 2 > w I and E , # 0 ; Fix an arbitrary bipartite partition of G'' and partition HI accordingly;

1 1

+

I

W.-L.Hsu,G.L. Nemhauser

362

end: else begin comment from linear programming duality there exists a clique in N ( u ) containing A l and one end point of each edge in E,. Hence the vertex set of HI can be partitioned into Vi and V2 such that all edges of Hi have o n e end in V, and the other in V2, and A I C V,. Furthermore, Vl is an independent set in G"; the procedure for constructing this partition, called PART(A,, H I ,G"), is given in the Appendix; PART(A,, HI,G"); let the LHS (left-hand side) of HI be V , ; end ; 514--W,,,-max{wt(v)I ~ P L H Sof H , } ; s1+w,,, - max{wt((x, y ) ) (x, y ) E E(HI)but {x, y } n (LHS of HI)= 81; s * +-min{sl,sz}; let C be the clique (LHS of H I ) U { u } in G with weight s * ; C,+C, U{C}; y c + - s * H,, + H , , / C ; wt(u)+-wt(u)-s*; for each u of H , in C \ { u } do w t ( u ) + w t ( u ) - s * ; for each ( x , y ) of E ( H . ) with either x or y in C do

1

wt((x, y )) + wt((x9 y 1) - s *; end; end; comment C,, is now the output of the algorithm; end.

Theorem 1. The algorithm outputs a ,finite collection of cliques {CL}such that P\!{U} is a maximum weight packing in the graph reduced by these cliques. Furthermore. yc;, = wt( u ).

z,

Proof. By definition, w,,,+wt(P\{u}) is the weight of a maximum weight packing in G that does not contain vertex u. Since P is a maximum weight packing in G, we have wmsrs wt(u). Consider two cases: (1) w,,,,< wt(u). Let C be the clique { u } in G with weight (wt(u)- wmaX). Clearly P still is a maximum weighted packing in GIC since now wt(u) = w,,, . (2) w,,,.,~ = wt(u). The optimality conditions (1) and (2) are satisfied in the reduced graph by the construction of C with weight y , = s*.

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363

To show the finiteness of the algorithm, we observe that s * > 0 and at each clique addition into C., one of the following will happen: (i) The weight of u will be reduced so that A l U A2# 0 in H,, (case (1)). (ii) At least one more vertex u of H, with wt(u) > 0 will be added to A l in the next iteration (when s * = sl in case (2)). It should be noted that once a vertex u is added to A 1 it will remain there until wt(u) is reduced to zero (since wt(u) = wt(u) at that stage). (iii) At least one more edge (x, y ) of H,, with wt((x, y)) > 0 will be added to E2 in the next iteration (when s * = s2 in case (2)). Again, once an edge (x, y ) is added to E2it will remain there until wt(u) is reduced to zero (since wt((x, y)) = wt(u) at that stage). Finally, once all vertices u of H , with wt(u) > 0 are added t a A l and all edges (x, y ) of H,, with wt((x, y)) > 0 are added to E2, then s * = s1 = sz = w,,,.~= wt(u). One more deletion of a clique with weight s * will reduce wt(u) to zero. Hence the number of iterations ( = number of Cl's) is bounded by O(l V(Ht)I+ IE(H'$I). Since W(u) is reduced by s * > O at each step, we have x,yc: = wt(u). Consider the efficiency of the algorithm. Since the final clique collection is composed of a set { C. u E P}, we only have to consider the construction of each C,,. The construction of H,, requires at most O(l V(d'))I') calculations of edge weights and lower vertex weights. Each such calculation requires an application of Minty's weighted packing algorithm, which is polynomially bounded. By Theorem 1, the number of iterations in constructing C,,is bounded by O(1 V(HO.))+ IE(H;)I). The procedure PART takes no more than O(l V(H.)21) time; the calculation of sI,s2 and s * takes at most O(max(1 V(Hu)l,lE(Hy)l)) time. Therefore, the entire algorithm is polynomially bounded.

1

4. An algorithm for minimum cardinality colorings on claw-free perfect graphs* Let f be a function mapping each vertex of G to a color. f is said to be a coloring of G if for any two vertices x and y, (x, y ) E E j f ( x ) # f ( y ) . Now consider a coloring f of G. Let i, j be two colors in f and consider the subgraph Gij induced on vertices colored i or j . An (i,j>path in G is a simple path with vertices colored alternately in i and j such that no two non-consecutive vertices are adjacent. Note that such a path may be a simple cycle.

R o p i t i o n 1. Each component of Gi, is either a single vertex or an ( i , j>parh. Actually, the presentation of this section only assumes that our graphs are clawdrtc and without odd holes or odd anti-holes.The algorithm gives yet another proof that this classof graphs is perfect.

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364

Roof. It suffices to show that each vertex in a component of Gij has at most two neighbors. Suppose not, and assume without loss of generality that y has at least three neighbors and f (y ) = i. Then since f is a coloring, all the neighbors of y are colored j and form an independent set. But then y and its neighbors contain at least one claw centered at y.

By switching of colors in an ( i , j>component, we mean changing all i's to j and all j ' s to i. Note that such a switching will still result in a coloring of G. Hence if x and y do not have the same color and are in distinct (f(x),f(y))-components, then we can switch the colors in one component so that x and y will have the same color. We use the fact that the number of colors in a minimum cardinality coloring is equal to the size of a maximum clique, w ( G ) , which can be found by the algorithm of Section 2. The algorithm colors the vertices successively to maintain a coloring that uses no more than o(G)colors. Thus a vertex is assigned an arbitrary color different from those of its already colored neighbors so long as these neighbors have used fewer than w ( G ) colors. However, if these neighbors have already used w ( G ) colors then we switch colors along appropriate (i,j)-paths. These paths are identified from the solution of an edge-matching problem as described below. Suppose that in a coloring of G \{u), N ( u ) uses w ( G ) colors. As in Section 2, let Gobe the bipartite graph which is the complement of the graph G" induced on N ( u ) . Since G is claw-free, at most two vertices in N ( u ) can have the same color. Let A4 be the set of edges in between two vertices with the same color. Then M is an edge-matching in G".

c"

kmma 2. If N ( u ) uses w ( G ) colors then M is nor a maximum cardinality matching in

c".

Roof. Let K be the set of colors that color two vertices of N ( u ) . Let

I

V, = { u u E N ( u ) , f ( u ) E K } . Then IV,I=21KI and J N ( U ) \ V I I = W ( G ) - I KHence J.

I N(u ) I = 1 Vi I 1 N ( u ) \ Vi I = 2 I K I + f

=w(G)+

0

(G ) - I K

1

I K 1.

Since Guis bipartite and the size of a maximum packing S in is =sw ( G ) - 1 , by Konig's theorem, the size of a maximum matching in G" is equal to IN(u)l-

Is 123 4 G ) + IK I - ( w ( G ) = IKl+

1.

1)

Algorithms

But IM 1 = 1 K

1.

Hence M is not maximum.

365

0

By Lemma 2 we have an augmenting path P in G" with respect to the matching M. O u r theorems show how this augmenting path identifies (i,;)-paths for color switching so that the number of colors for N ( u ) is reduced by one.

Theorem 2. If P consists of a single edge ( x ,y ) where f ( x ) = i and f ( y ) = j , then by switching colors along the (i,j)-path in G containing y and assigning vertex u the color j , we obtain a coloring for G that uses w ( G ) colors. Proof. It suffices to show that the (i,j)-path in G containing y does not contain x. Since (x, y ) kZ E ( G ) , an (i,j)-path joining x to y must contain at least two intermediate vertices. Since ( x , y ) $Z M, none of the intermediate vertices is in N ( u ) . But this implies that the cycle formed by the (i,j)-path together with the two edges ( u , x ) and ( u , y ) is an odd-hole. 0 Theorem 3. Let P = ( x o y l x l y *z* . xlyl+I' . . x,yo) be an augmenting path with respect to M with a 3 1 and suppose that P cannot be shortened by the existence of an edge ( x i , y m ) , 13 1 , m > 1 + 1 . Then by switching colors along a n (i,j)-path we can obtain a different coloring of G \ { u } with a corresponding matching M ' and augmenting path P' such that P' is shorter than P.

A constructive proof of this theorem is given in [6]. Here we will just show the color switches that we require without justifying that they are valid. Three types of switches are required as shown in Figs. 1 , 2 and 3 . In each case the figure gives the relevant part of Gowith matching edges indicated by heavy lines. Fig. 1 is for a = 1. It can be shown that there does not exist an (i,j)-path joining xo and y , and an (i, k)-path joining x I and yo. It there is no (i,j)-path joining xu and y l , then by switching colors along an (i,j)-path containing y l and revising the matching accordingly, we obtain the augmenting path consisting of the single edge (yo, x l ) .

Fig. 1.

W.-L.Hsu, G.L. Nemhauser

366

Figs. 2 and 3 are for a > 1. The vertex coloring on P is given by f ( x o ) = j , f ( y o ) = k and XI) = f ( y r ) = irr 1 sz I S a. In Fig. 2 it is assumed that there exists t, 1 g t S a, such that ( y I , x l + l ) EE ( c o ) . It can be shown that the ( i l , il+l)-path containing ( y l , y I c l )contains neither xI nor x,+~.Thus by switching colors along this (it, I+l)-path and revising the matching accordingly, we obtain the shorter augmenting path through x , - ~y , ~ yI~ +2-+ ~

@

Yt+2 it+2

i t r 1 Xt+1

Yt.1

it

4 - i1t xt-1 xt

Yt

it.1

W Fig. 2.

Fig. 3.

In the final case, as shown in Fig. 3, it is assumed that Vt, 1 S 1 G a, ( y , , x ~ +E~ E ) ( G )so that the switch shown in Fig. 2 is not possible; furthermore, it is assumed that there is an (il,j)-path 0 from y r through x 1 and X o so that setting f ( y , ) = j as in Fig. 1 is also not permissible. Then it can be shown that the colors can be switched as follows: f ( X o ) = f ( y l ) = i2, f ( y 2 )= j , f ( x z ) = il, and switch colors on the (il,j)-path Q. The new matching yields an augmenting path ( x 2 y 3 . . x,yo) that has four fewer edges than P. The inductive application of these color changes ultimately yields an augmenting path of length one so that Theorem 2 can be used to color u. The running time of the algorithm is determined by (1) the number of iterations, which is O ( n ) ; (2) finding an augmenting path in the complement graph of a neighbor of a vertex, which can be done in O(n2')time, see [ 5 ] ; (3) shortening an augmenting path, which is done by testing for the existence of certain (i,j)-paths in G", which can be done in O ( n 2 )time; the number of shortening is at most O ( n ) .

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367

At each iteration we might have to do both (2) and (3), which can be bounded by O ( n ’ ) . Hence the entire algorithm is bounded by O(n4).

Appendix

To find an appropriate partition of the vertices of HI in the case of A , # 0, our algorithm uses the procedure PART(A, H, K ) where A is an independent set of K ; H = ( VH;E H )and K = ( VK;EK)are bipartite graphs such that VH= V K , EH C EK and the vertex set of H can be partitioned into V1and V 2such that all edges of H have one end in V1 and the other in Vz, and A C Vl. Furthermore, Vl is an independent set in K. The procedure PART will make a correct bipartite partion of the vertex set of H and output V1 (the LHS of H ) . procedure PART(A, H, K ) : begin 1 V, t { u E VH there is an even length path in H leading from v to some u €A}; 2 Q + a list of components of H which have not been partitioned; 3 I=1; while I# 0 do begin 4 I=O; for each component B in Q do 5 begin if there exists an edge (v, x ) in K connecting v E V1 to x E B 6 then do begin comment the vertex x must be placed on the RHS of H ; 7 delete B from Q ; comment it is easy to determine a partition of a connected bipartite graph according to whether the length of a path between two vertices is even or odd; arrange the vertex set of B such that x is on the RHS; 8 if (LHS of B ) # 0 then do 9 begin V1+ VI U (LHS of B ) ; 10 11 f t l ; end end

1

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W.- L.Hsu, G.L. Nemhauser

else comment restore this component to Q for future consideration; end comment ‘ I = 0 indicates that V, has not been augmented in this loop; end 12 K ’ t t h e subgraph of K induced on U Q ; 13 Make an arbitrary bipartite partition of K’; 14 Vi + Vi U (LHS of K ‘ ) ; end The correctness of the algorithm can be argued as follows. In line I we have that A C V, and VI is an independent set in K. Each time VI is augmented in line 10, the LHS of B is independent of the old V, since otherwise there cannot exist a correct partition. In line 14 no vertex of K’ is adjacent to any vertex of the old V,. Hence the final V, is an independent set in K . Let m be the initial size of Q. Then we have to execute line 6 through line 11 at most m ( m + 1)/2 times. Therefore, this algorithm is polynomially bounded.

References \ I ] V . Chvdtai. On certain polytopes associated with graphs, J. Comb. Theory, Ser. B. 18 (1972) 13S154. [2] D.R. Fulkerson. On the perfect graph theorem, in: T.C. Hu and S.P. Robinson, eds., Mathematical Programming (Academic Press, New York, 1973). [3] R. Giles and L.E. Trotter, On stable set polyhedra for K,,3-freegraphs, J. Comb. Theory, Ser. B 31 (1981) 313-326. (41 M. Grotschel, L. Lovasz and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization, Combinatorica 1 (1981) 169-197. [S]J.E. Hopcroft and R.M. Karp, An nS’* algorithm for maximum matching in bipartite graphs, SIAM J. Comput. 2 (1973) 225-231. 161 W.-L. Hsu, How to color claw-free perfect graphs, Ann. Discrete Math. 11 (1981) 189-197. 171 W.-L. Hsu. Efficient algorithms for some packing and covering problems on graphs, Ph.D. Dissertation, School of Operations Research and Industrial Engineering (Cornell University, Ithaca, New York. 1980). 181 W.-L. Hsu and G.L. Nemhauser, A polynomial algorithm for the minimum weighted clique cover problem on claw-free perfect graphs, Discrete Math. 38 (1982) 65-71. [Y] W.-L. Hsu and G.L. Nemhauser. Algorithms for minimum colorings by cliques and maximum cliques on claw-free perfect graphs, Discrete Math. 37 (1981) 181-191. 1101 A.V. Karzanov, Determining the maximal Row in a network by the method of preflow, Soviet Math. Dokl. 15 (1974) 434-437. [ 111 L.G. Khachian, A polynomial algorithm in linear programming, Soviet Math. Dokl. 20 (1979) 191-194

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(I21 G.J.Minty, On maximal independent sets of vertices in claw-free graphs, J. Comb. Theory, Ser. B 28 (1980) 284-304. [13] K.R. Parthasarathy and G. Ravindra, The strong perfect graph conjecture is true for K,,,-free graphs, J. Comb. Theory, Ser. B 21 (1976) 212-223. [14] N. Sbihi, Algorithmes de recherche d'un stable de cardinalitt maximum dans un graphe sans ttoile, Discrete Math. 29 (1980) 53-76. 1151 I. Holyer, The NP-completeness of some edge-partition problems, SIAM J. Comput. 10 (1981) 7 13-7 17.


Topics on Perfect Graphs (Annals of Discrete Mathematics, Volume 21)

Topics on perfect graphs

Studies on Graphs and Discrete Programming (Annals of Discrete Mathematics)