Handbook of Graph Theory

DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN HANDBOOK OF GRAPH THEORY EDITED BY JONATHAN ...

Author: Jonathan L. Gross | Jay Yellen

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DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN

HANDBOOK OF

GRAPH THEORY EDITED BY

JONATHAN L. GROSS JAY YELLEN

CRC PR E S S Boca Raton London New York Washington, D.C.

© 2004 CRC Press LLC

DISCRETE MATHEMATICS and ITS APPLICATIONS Series Editor

Kenneth H. Rosen, Ph.D. AT&T Laboratories Middletown, New Jersey

Charles J. Colbourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs Charalambos A. Charalambides, Enumerative Combinatorics Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses, Constructions, and Existence Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information Theory and Data Compression Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and Nonorientable Surfaces Richard E. Klima, Ernest Stitzinger, and Neil P. Sigmon, Abstract Algebra Applications with Maple Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science and Engineering Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration and Search Charles C. Lindner and Christopher A. Rodgers, Design Theory Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied Cryptography Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Fundamental Number Theory with Applications Richard A. Mollin, An Introduction to Crytography Richard A. Mollin, Quadratics


Continued Titles Richard A. Mollin, RSA and Public-Key Cryptography Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary Approach Douglas R. Stinson, Cryptography: Theory and Practice, Second Edition Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and Coding Design Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography


8522 disclaimer.fm Page 1 Tuesday, November 4, 2003 12:31 PM

Library of Congress Cataloging-in-Publication Data Handbook of graph theory / editors-in-chief, Jonathan L. Gross, Jay Yellen. p. cm. — (Discrete mathematics and its applications) Includes bibliographical references and index. ISBN 1-58488-090-2 (alk. paper) 1. Graph theory—Handbooks, manuals, etc. I. Gross, Jonathan L. II. Yellen, Jay. QA166.H36 2003 511'.5—dc22

2003065270

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microÞlming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of speciÞc clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-58488-090-2/04/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. SpeciÞc permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identiÞcation and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2004 CRC Press LLC No claim to original U.S. Government works International Standard Book Number 1-58488-090-2 Library of Congress Card Number 2003065270 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper


PREFACE

! " "

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% & Format

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1 ( " Terminology and Notations

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9 ( ) : Æ 6 7 * ( ;

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Acknowledgements

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References 5A;7 " " H/4 " CA; 5CA7 #I "

" " CCA 5?C@7 ? ?!" " - " 4" CC@ +2 - " J= J#" C:C,

© 2004 by CRC Press LLC

Section 1.1

Fundamentals of Graph Theory

19

5DCC7 K ? K D"

" J" CCC 54C7 2 4"

" J " CC +2 - " % /=" C@C, 5 C 7 L 8 H "

" K = M " CC 5667 = "

" # " 666 5=6 7 ="

" - " J/4" 66 +2 - " CC@,


20

Chapter 1

INTRODUCTION TO GRAPHS


1.2

FAMILIES OF GRAPHS AND DIGRAPHS

! ' /

/ . ; ; DEFINITIONS

) % # + , " & +2 " &½ , ' , $ " # ,$


Section 1.2

Families of Graphs and Digraphs

25

) % # . & ' , $" &

) % ) # ) +

, &

) % ) )/ & > % ' ) ½ " &½ ¾

" ," & EXAMPLES

) )

- #

- # #" # #

)

/ & /

Figure 1.2.7

&" +"" & . *

FACTS

(

) 5 3 7 % # + " 5DCC" ;7,

( ) % )/# )/ ( ) 2 ) " # )/ HJ/ k-Connectivity and k-Edge-Connectivity

# 3 $$ *$$ : . DEFINITIONS

) " - + ," # /

) " - + , # /


26

Chapter 1


# 0 1 - " - . - - "

)

% - ) ) -9" )/ ) /

) % / - ) " ) )/ /

) % ) + ) , )

) % ) + ) , ) Minimum Genus

# 3 DEFINITIONS

) + , # # # / + : , /

)

% 6

1.2.4 Criterion Qualification % " " # . DEFINITIONS

) % ! . + . ,

) % + ; . ,

) % )/ ) # + ; ,

) % )/ ) . + ,

/

) % )/ / ) // + ,

) 0+ , B 0+ , & . %" # . 0+ ,


Section 1.2

27

Families of Graphs and Digraphs

) % . # + Ü, EXAMPLE

) Figure 1.2.8

"

FACTS

() ¯

5? 37 9)

¯ 5L7 # # . ¯ 5:67 H 2 C #

Figure 1.2.9

()

!& !

%

EXAMPLE

)

Figure 1.2.10


"!" %" " ! '"&

28

Chapter 1


References 5:67 ? = !" 3 "

C + C:6,"

CP ; 5DCC7 K ? K D"

" J" CCC 5L7 K L3" Q R IQ = I." ;6 + C," :;PAC 5=CA7 K =" ! " (. F J" CCA


Section 1.3

1.3

29

History of Graph Theory

HISTORY OF GRAPH THEORY

!" #

# ; %

Introduction % A:A" / # #! :; ? - + :6:/A, LS # # # 3 2 # 5?=CA7 5=CC7

1.3.1 Traversability # #! -T ! LS # # # + :;," #9 " # J L! + A6@/C;, = 4 + A6;/@;, " The Königsberg Bridges Problem

#. " " 2 " ! / ! # LS # . O " ! # Ü . " C

c

d g

A

c

D

a B

Figure 1.3.1


b

' 78

30

Chapter 1


FACTS 5?=CA" 7

() ( @ % :; ? - 0 #/ 1 % J# " " ! # () ' :@" - " / # () - 5-) :@7 # " # 0 # / 1 % :@" : " # :; () -T " # #"

LS # #

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() - " # "

#" # " # % # 43 54) A:7 A: Diagram-Tracing Puzzles

% * $ "%% " 9 # # ! 33 # #! P ." % .


Section 1.3

31


FACTS 5?=CA" 7

( ) ' A6C ? J 5J) A6C7 #)

" 9 $.# # # " & 9 J # # " # ' " 9

( ) ( / 33 # 5) A7 K ? 5?) A:7 #! $ " 0 1 () ' AC" ( 9 ! # # # " # 8 5) A: /7 #

&

() # LS # # # /

33 3 C ' # = = 5) AC 7

% # # 2 # C g

c d e

A b

D f

a B

Figure 1.3.2

" 78

Hamiltonian Graphs

% # # # & . 4T $" " + Ü;," & # L!" # 4T " # FACTS 5?=CA" 7

() % . # / " " ! T # " @ 9 & # # #!


32

Chapter 1

" 5 2 A H

H

H

H

H

H

C

C

C

C

H

H

H

H

H H

C

H

C H

H

H

C H

Figure 1.3.8


C H

H

H

% > !" !"

Section 1.3

35


() 5) A:7 / Æ +!,

# " B # # ! * A 2 H#

;

C

A

() = L > # # #

# 9 " # ' A:A" 5) A::/A7 & # " ! ) - # .# # " L!I

" /

( ) ' A:A" 5) A:A7 !

" ( ) ?

C 6 C6 % ? K L 5? C7 3 " JI T #/ #

1.3.3 Topological Graphs -T 5-) :;67 " ' . ' C6" 3 # # # L33 L ! + AC@/ CA6," ! P # # H #" J " P . Euler’s Polyhedron Formula

! " # ! # # " " 1 )

G1 *

' : " I " # -T # 4 " " # !


36

Chapter 1


FACTS 5?=CA" ;7 5CC7

() " H# :;6" - # # " +, & + , # " " " ' # # # . # # " G * G

() - # ' :; # / " # # %/8 ? 5?) :C7 :C" () ' A " %/? 5) A 7 # -T # /

&

() % " /%/K ? 5?) A 7

. " # 4 -T P " " / + " RT , 2 / " ?

G1 * 6 . " # "

G 1 * # & &$" 9 1 $ $ $ + : ,

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( " ./

." > 3 # -T ! # $ #9 ' " 4 JI ! ? T AC;/ C6 #

() ! JIT ! " /

# 8 J 4 5 46:7 + , / ) '

* ! 4 # ( + A@ / C@ , # " # 4 3 + AA6/ C@, /# " # # # D 4 & C@6 H # J 3 L !T CA6 FACTS 5?=CA" :B :7

() ' AC6" 4 54) AC67 #

&

4 # # " #

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# " 4 T #

() ' C 6" 4 3 5 67 . 4>T // # " 8S # # & " / 4 4 # L #" # # J 2! 527 C" . ' C;" ' L 5L;7 /# " " @ () 4 /# C; #

# # Æ" 66 8 / C@6" C@A # D 5D@A7" = T 5@7 # C@ $ " " # K ? 5:7 3 # #&" 3 " + Ü:," + = ! 5:7,

() ' 9 CA6 " #

5A;7 " # " 0# # 1 + Ü::, 4 " " # # # " 2 /# " " C:C 4 4 " K J 4!" = 54=:C7 # 6 # # &


Section 1.3


39

1.3.4 Graph Colorings - ! " " # / # # 2 A; " # +, 01 # % L A:C ' # L % = 4! C:@" # ! L" !>" 4 4" " #9 # H #" " J " # 5 CC7 8 " # " 5 AA67" 5 7 # ) " )" $ $ "

() ' C

" 2! 52

7 # # / " / #

;

() % C;6 4 # # / ( . " 4 54@C7 #


Section 1.3

41


() ' C:@" % 4! 5%4::" %4L::7" K L"

# # A # " # / 9 # / #

() % CC" #" " " 5C:7 /

F # # # " 3 %/ 4! " # # @ #

Other Graph Coloring Problems

% ! / #" #

#

FACTS 5?=CA" @7 52=::7 5KC;7

( ) ' A:C " L 5L) A:C7 # & > / ! # 4 = C # #9 ! / #

( ) ' AA6" 5) A:A/A67 / 9 # .

() ' C @" LS 5LS @7 # . #

+ Ü

,

() & > C6" ! =" J () ' C " ? ! 5 7 #

. G " + Ü; ,

9

() ' C;6" # ./ # % " $ $ "

() ' C@" " % # " ! + C ," P@

5 7 !>" # " ; + C ,"

;P A

5?@7 !> ? " " " @6 + C@," ;;P; 5 @7 ( !" ( &I #I I I" " " & 2 + C @," :P;A 5 7 ? !" ( !" # # " : + C ," CP C: 5) A 7 %/? " Q / I" ) #0 C + @, + A ," @APA@ 5) A;:7 % " ( " # +, + A;:," : P :@ 5) A:7 % " ( " # +, : + A:," P@ 5) A:C7 % " ( " # % " + + A:C," ;CP @

,

5) AAC7 % " % " + # + AAC," :@P :A 5: 7 % !" . / " ; P ;A # . "

! " %8" H D!" C: 5) A7 " 5 7 #" & + A," 6CP @ 5CC7 J " #

" # F J" CCC 5 A 7 3 " # " 3 % + CA ," PA 5 2K;7 3 " 2! 8 K" / / #" 3 % + C;," CP 6 5 46:7 8 J 4 " % " ) '

* ! + C6:," ;P 6 5 8) A@67 % 8 " % " " # = = " "

H @C + A@6," ;6 P;6


Section 1.3

45


5 ;C7 - = &!" % # . " & + C;C," @CP : 5 ; 7 % " # " # " +, + C; ," @CPA 5 7 4 - " J." " @ +K C ," C ,"

6 +%

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!"

52

7 J 2!" #" + C

,"

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! � " = 4 2 " C:C 54F=:C7 4 4 " K J 4! = " 6 # & "

+, : + C:C," P:6 54@ 7 - 4" 8/ ! $ " " C + C@ ," ;; P;;@ 5:7" K ? " % $" " "

&" P ) . # &&" ) # ! )+&&" ) ) " " #

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52

Chapter 1


* + ,) ./ " # /

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53

Chapter 1 Glossary

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54

Chapter 1


1 ) "

$ " !"+&) !"+ ) " !" ) / . " # /

P .) & . " P ' ) ' ! ) ./ / B &¼ &" ) / / &¾ " '"1 P )

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55

Chapter 1 Glossary

+) & +!. + P ) & % *

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%)$ &&" ) " % " P ) + $ , #


Chapter

2

GRAPH REPRESENTATION 2.1

COMPUTER REPRESENTATIONS OF GRAPHS

2.2

THE GRAPH ISOMORPHISM PROBLEM

2.3

2.4

THE RECONSTRUCTION PROBLEM

RECURSIVELY CONSTRUCTED GRAPHS

! ! " " " GLOSSARY


Section 2.1

2.1

57

Computer Representations of Graphs

COMPUTER REPRESENTATIONS OF GRAPHS

Introduction

! "

# # $ % #

2.1.1 The Basic Representations for Graphs # & ' & DEFINITIONS

(

(

) * + ! ,

- * +

) * + ! , -

( . ) * +, ' ' * + & ( . ) * +, ' ' & ( / *½ ¾+, *¾ ¿+,, * ½ + ' ½ ' ( ) * + # ¾ ( ) * + # ¾


58

Chapter 2 GRAPH REPRESENTATION

( ) * + ' , # 0 1 ) ' ' 2 0 1 ) 3 #

(

) * +

, ' 4 ' , % & %

(

' , # ' 0 1 ) ' 3 #

4 , 0 1 )

) * +

3 #

EXAMPLES

( 4 # & ' &

! "# # ! !

Figure 2.1.1

(

' ' # #

)

* +

3 3

* + * + * + * + 3 3 3 3 3 3 3 3

FACTS

$(

& '

$(

) * + # % *

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*

5

¾+

+

Section 2.1

59


REMARKS

%

(

4 , 06781, 0

6791, 0:;

¾

) *

/

+ , & ' &

*

+ 6# , # & ', #

# ' , # # & #

*

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#

%

(

? &

# & ( '

,

'

2.1.2 Graph Traversal Algorithms @ % A ! ! /

/

4

/ , & # % #

Depth-First Search ALGORITHM A ! . , % B#C > ' , % BC A ! # % # ' &

, % ,

# B ! C

# * + % #

DEFINITIONS A , !

%

, # (

(

(

*

*

+ #

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+

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!

(

*

+ '

!

(


60


&$# '

Algorithm 2.1.1:

# ( ) * +, # ) 01 & ' $ # ( !

!( *+ ! () ! !

01 () 2

! () ! ! 01 ) 2 *+2

!( *+ 01 () 2

! ' 01 ! 01 ) *+2

FACTS

$( $(

+ ) * + . # ! ' # B*C B+C, / ! A ! %

*

5

' #

$(

. ! , % REMARKS

%

( A ! =D3 08 681 Æ !

%

( A ! !

Breadth-First Search

! / ' . ! ' # . !

' ,


Section 2.1

61


ALGORITHM

! ! ! ! ! * "+ ' % / " ! ! *"+ / " !

, % 5 . , ! % % Algorithm 2.1.2:

) &$# '

# ( ) * +, # ) , 01 & ' , ! ' $ # ( !

!( # * + ! () ! !

01 () 2 01 () 2 01 () "2

01 () 2 01 () 32 " / "2

! ! * "+2 *" " ! () ! ! *"+2 ! ' 01 ! 01 ) 01 () 2 01 () 01 5 2 01 () 2

! ! * "+2

DEFINITION

"

( ; #$ # , 01 , * 01 + 01 #$ # * +

"

FACTS

$(

! %


*

5

+ ) * +

62

# * + 01

$(


REMARKS

%( ; % ! , ! =D3 -

! " #

%( ! % A &% E E !

2.1.3 All-Pairs Problems # ( #

4 & '

All-Pairs Shortest-Paths Algorithm

< # # # # # %# %# > # A &% E ALGORITHM

# 4> # 4> & ' # ) * + 4 # ' % 0 1 * + . % 0 1, # % 0 1 ! >

4>

, # 0 1

' ' 4 # %

, FACT

$( 4> ' # ) * + * ¿+ * ¾+ REMARKS

% ( 4 4> 06781 0:; 4, =8 *8=1, > , *+ @ ! , . & $ DD9, < , =8G 078G1 K 7, , %# $ *=8G+, J


Section 2.3

2.3

79

The Reconstruction Problem

THE RECONSTRUCTION PROBLEM

# :& * + ! * +

#

# % * + ) * + * +

>

# !

$( $( $ ( $ (

6 # !

The Characteristic and the Chromatic Polynomials DEFINITION

( #

4

, * +

* +

FACTS

$(

0 ; F , @ , *=9+, 3J F9D1 > ; F , @ , % E N K 7 981+

$( $( $(

# # & %%

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Figure 3.2.4 && %'

2 = > ! #

" # #

!

! ! 6 # 6 # B #"! # )! # ! " # "

&

2 $ % " !

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2 7" ) 6 2 7" ) 4 ! !

2 7" ) ) 2 " ) ) 2 ) " # 6 2 ) " !

1 !

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7 # " - " " = -> 6 = " "> " / H #0 "! 6 < !

- /! $&'(0 DEFINITIONS

2 " - -

2 "

- -

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2 % ! % ! " " 6 % " 6 FACTS

2

$G C( 7" 6 3 ! " " 6 $% C (

2 %

$ H,( " % ! - # 6 % ! % 5 ! /% 0 5 /9 90 / $H(0


Section 3.3

Tournaments

169

2 $H,( , % ! - #

6! ! " 6

# A2 /0 % ? ! /0 5 % 5 ! / 0 5 % 5 5 9 % ! ! ! /90 / % 0 / 9 0! /C 9 ,0! /: ' 0 / $ & ( 0 %

2 $H9( %- " 9% ? " "!

# 4" 2 /0 %- /9% ? 0 /0 /9% ? 90 /9% ? 90 6#- B / 0

/9% ? 0- 2 $ H,( 3

" " 6 / $ 'H(0 3 ! " %!

%- $H9( REMARKS

2 " 1 -

# 6 " # $H( $+ H9( ) #

" ) 6 - # $H,( # 6 6 $&'(

2 A ! 6 ! Tournaments Containing Oriented Trees

/ 0 ! / 0 - #

" CONJECTURE

7 * + / $+ H (02 7" / 0- " " EXAMPLE

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223

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229

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230

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Section 4.2

231

Eulerian Graphs

4.2.6 Transforming Eulerian Tours The Kappa Transformations 6 &

6

( -(/10

DEFINITIONS

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234

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- (/40 $ * ( , + =//4> 9K -;/0 2 ; ;

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A F 2 8 5 % %8 =/4> 1K9 -A970 : A =97> 9 3K 1 ? ; " L 9 K1 -(330 ( .# + 33 49K7

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Section 4.2

Eulerian Graphs

235

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%

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236

Chapter 4


-&30 . J & A

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238

Chapter 4


The Eulerian Case DEFINITIONS

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239

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241


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242

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243

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244

Chapter 4


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Section 4.3

245


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246

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248

Chapter 4


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Section 4.3

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251

References -889 0 A : 8 : . 8 F!& L $ + '. =/9 > 74K/ -8 6/0 8 8 & 6 . . 2 %($ + $ =//> 3K41 -8 6/0 8 8 & 6 2 : 6 . 2 ( 9 =//> 444K43 -8310 8& 6 $ # F!& ( 9 -A74 0 H A$ # $ ! 1 L + ) ' , % 7/8 =/74> 4K1 -A74 0 H A 6 (! + $ 9 =/74> /K 79 -AH90 H A A H $ A 6 $ 4 =/9> 33K -A840 6

A F 2 8 6 ; : 2 % %8 3 =/4> 1K9 -A 2: /4 0 A $ 2 2 : "' 6 5 ) =//4> K -A 2: /4 0 A $ 2 2 : ""' 6 5 ) =//4> //K -(/0 * ( A 2 6 L $ 41 F * =//> -((70 : ( . (& -. '. N FH =/7>


252

Chapter 4


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Section 4.4

4.4

253

DeBruijn Graphs and Sequences

DEBRUIJN GRAPHS AND SEQUENCES $ %&' ( ' .8 5 2 8 2 8 5 B F 2

Introduction F 8 5 , = > ( ! 8 5 " ! 8 5 ! 8 5 B

4.4.1 DeBruijn Graph Basics DeBruijn Sequences DEFINITIONS

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Section 4.4

255

DeBruijn Graphs and Sequences 0000

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1000 1001

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256

Chapter 4


FACTS

' -2 70' ! # A ' " 8 5 => B #

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' ( 8 5 B 8 5

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Section 4.4

257


Necklaces and Lyndon Words

( & J -(J990 & 8 5 B DEFINITIONS

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>

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258

Chapter 4


'

6 : ! 11 1 + 1111 8 5 B

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4.4.3 Pseudorandom Numbers ( $ = >

@ 8 $ B .8 5 B ! ! DEFINITIONS

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6 %

@ 1%

B 6 ! 5 = E > ! ! = E 11 11>

' B ! ' 1 2 8 5 B # 1% # %

'

-390 6 .

+

,

= >

7 / /

! 7=/> 6 , / B +


Section 4.4

259


FACTS

'

; 8 5 B

B

'

8 5 B , 2% , ! (

% # B 1% !

B #

'

" = # > 8 5 B

&

;

B 2 5 8 5 B ! ! B

'

-H/0' 6 !

B

8 5 B

4.4.4 A Genetics Application 6 .F # Æ 8 # # !& 6

.F ! 6 #

, B # ! .F B B #

6 Æ 6

+ - 6 + 10 , 8 5 !

B 7

B

DEFINITIONS

!)

'

'

( + ?

, B 2 6

½

7 7¾ 7

.F B ! , +

+

6!

5 .F B 7 ! ,

REMARK

'

A B & +

8 5 Æ !

6 = > B

! # ! @ + 8 5


260

Chapter 4


References -;0 " & & : ! 8 5 B

!!! M M M&MF& M -390 2 H ( N /39 -. 70 F 2 8 5 ' 943K97

/

-(J990 * ( & " J :# 8 5 B + % =/99> 9K1 -H10 H H ; B /7K1 8 * B!A = : 8 . > L 11 -2 H9/0 $ 2 . H ( 1 ' + * ( X /9/ -2790 + 2 % ) % * . /79 -2 70 " H 2 F + 0 $ % =/ 7> 79K 9 -22$ ://0 H ( 2 Æ + $ 2 H * $ :! $ + * ( /// -2


Section 4.5

4.5

261

Hamiltonian Graphs

HAMILTONIAN GRAPHS '

' )*

4 * 4 6 & 4 A# 4 $ 6 ; * T 44 2 47 (

4.5.1 History F = -2 H9/0> ! F + ! * 341 * # # *

. 349 6 ,

6 ! & ! 34/ ! , ! * % 4 * , B " -J 470 344 6 J & B ' 2 ! , => # 6 J & & B * N J & ! ( -8 :+ 370 DEFINITIONS

' ' '

= >

4.5.2 The Classic Attacks 6 @ 6 & Æ U #


262

Chapter 4


6 # ( Degrees

6 # Æ = >

Æ =

>

DEFINITIONS

' + = > ! 6 = >

' 6 ! = > 5 5 !

' ( ' ? = 9 > = ? 9 > 5 5 ( 3 9 ! E

6 ! '

: =

>?

/ (

(½ (

FACTS

' '

-. 40 " :

-;710 " : = >

Æ = >

: = >

-;70 " := > E

EXAMPLE

'

! ! # , = ( 4> 6 * Æ = > ? =* > := > ? * . % 6 ;% 6 =( > 6 ! Æ = > ? * : = > ? * ! ; = ( 4>

Figure 4.5.1

' -H 310 :

2$$" ! " " & ;"& "$


!

Section 4.5

263

Hamiltonian Graphs

' -$$70 " ? = 9 > = > ! => E => E 5 9

'

-8970 :

! =

6

> ! ,

!½=

!=

>

>

' -*/0 REMARK

'

6 # 6 => & , *!

! " #

1 6 ! ! ! =! >

Other Counts DEFINITION

' 6 # ( - =(>

5 ( - =+ > + 5 # +

! "

+ ! ,

' 6 = > !

#

'

= >

EXAMPLE

' 6 =5 *> ! * # + ! + ? ? 5 ? * 5 ! ! 5 + * + Y ¾ ( 4 ! ! = 7> = 4> © 2004 by CRC Press LLC

264

Chapter 4

Figure 4.5.2

+"

= 7>


= 4>

FACTS

'

-;70 " ½ E ( ! # E

= > = 4> "

'

-( 3 0 "

/ / (=> => = > ?

' -8 8L: 3/0 " := > E = >

'

-A90 :

=

>

= > = > " = > = > " = > = > E ' -+930 " + - =+> ' -(370 : " # & + ! & ! - =+ > "

'

-8L/0 -( 2H :/0 "

- =+ > + ! REMARK

'

6 #

1

Powers and Line Graphs

! & = >

DEFINITIONS

' 6 ;= > ! ! ! ! ;= >

5 5 = >


Section 4.5

265

Hamiltonian Graphs

' !

#

!

' + 5 = ½ > 5 ' 6 ! = > ? = > ! = > = >

'

= # 5 > = >

"

FACTS

' -* F+740 : ! 6 ;= > ½

' -2*//0 : ! 6 ;= >

! = >

'

' '

-+ 90 " -(9 0 "

" -8930>

! Æ = >

; =

> ? ;=;= >>

= > =

Planar Graphs FACTS

'

-630 A = -6470>

' -2730 : ! ! " 5 ! 5¼ # ! = >=5 5¼ > ? 1

4.5.3 Extending the Classics Adding Toughness DEFINITION

' " # + + ! & &

, & =+ > + ! =+ > & 6 #

FACTS

'

-H930 :


:= >

6

266

Chapter 4

'

-8 $L/10 :


'

-8L/10 : 6

:= > 6

! Æ = >

REMARK

'

G 5 & & ( & ? *! -8 8: L110 # =/ 6> 6 $ 1 !

More Than Hamiltonian DEFINITIONS

' '

2 2

' ! / # / E =! > ( # #

2

' = > B = > B

FACTS

'

=> Æ = > => := > !

-8( 2:/90 "

EXAMPLE

' 6 ! 6 FACTS

'

-8990 "

'

! = > ¾

-*/10 " := > # ! : = > = 4> # ( Æ = > = E > #

'

-*/0 " ? = 9 >

5 ( 3 9 ! =(> E =3> E


Section 4.5

267

Hamiltonian Graphs

' -*/0 : / " ? = 9 > Æ = > / = > $ ¾ / E /¾

' -J /70 -J /30 6 # Æ = > = E > ! ' -J //0 : " => E # ' -( 2J: 0 : ! " => E => E = /> 5 REMARK

' 8 !

N# . ! ;

4.5.4 More Than One Hamiltonian Cycle? A Second Hamiltonian Cycle FACTS

' A 6 = -6 70>

' -6/30 "

/ ! / 11

! ! : # ! ! '* =! > = - => = =! >>> 6 ! ¼ ! ¼ ? ! # ! ! ¼ ! !

' -6/90 :

' -*110 ( # =>

! Æ = > => Æ = > Æ E " =Æ = >Æ = >>

' -$ 970 -2$ 970 6 #

! 5

!

Æ =

>

' -Q 970 -3/0 6 # , # 4

= > !


268

Chapter 4


REMARK

'

( 6 -6930 # % 5 ! 5 = ! > 6 # =( >

Many Hamiltonian Cycles FACTS

'

-6/70 :

! ' (½ 3½ (¾ 3¾ ( 3 (½

= > " 3½ 3 (½3½

=> " 3½ 3 $ ½ ¾ =1 ' > ! 3 ½ (3 ·½ = '>V ½

'

71¾ :¾ = > E

-( 340 :

= > "

5 => " 7 ¾ ½ E 5

' -A/0 : ! = > " ! :¾= > Æ = > 5

Uniquely Hamiltonian Graphs DEFINITION

'

#

FACTS

' -A!310 6 # , B !

'

-H +3/0 B # = E /> B #

'

-8H /30 A B #

¾ =3 > E ! ? = ¾ > ½ ( B !


Section 4.5

269

Hamiltonian Graphs

Products and Hamiltonian Decompositions DEFINITIONS

' = E >

'

A ! & # = ½> = ¾> 6 ? ½ ¾

=

> ? =½ ¾>=½ ¾> ½ ? ½ ¾¾

=

6

=

> ? =½ ¾>=½ ¾> ½½

?

=

¾> ½ ½>

¾ ? ¾

¾

½½ =

?

¾ ¾

=

¾>

¾ = > ? =½ ¾>=½ ¾> ½ ? ½ ¾¾ ¾ ¾ ? ¾ ½½ = ½ > ½ ½ = ½ > ¾¾ =

6

½>

½

¾>

6 = ! > ? ½- ¾0

=

> ? =½ ¾>=½ ¾ > ½ ½

=

½ ? ½ ¾¾ =

½>

¾>

REMARK

'

H & -H 9/0 5 E B ' " ½ ¾ ½ ¾ T

FACTS

' -/0 : ½ ¾ ! 7 & ! & 7 6 ½ ¾ ! '

7 & => & =>

=>

¾

= >

½

'

77&

" ½ ¾ * ½

¾

'

-8/10 -Q3/0 ½ ¾ " ½ ¾


270

Chapter 4


'

-( : /30 6 ½

'

-8 30 6 # !

' -J/90 = > "

=> =>

= > - 0 " = > - 0 " = > - 0

=> " =>

= > - 0

= > - 0 " = > - 0 " = E > = > E - 0 " # = > - 0

=> " =>

#

4.5.5 Random Graphs + 5= > ! - ? DEFINITIONS

'

=3 > 1 * : - ! * ' =3 . ( > 4 ? 4 = > ! &

6 7 ? @ ! 4 # + ! 7

' ! @ B = >

& & ? 1 -

' " Z ! Z " 5="> F B

"

' 6 4 2 %

# 2 6 %


Section 4.5

271

Hamiltonian Graphs

FACTS

'

- 970 -J970 6 # ¾ ' -J970 -J30 * ? E E 4 = > ? E E 6

'

4 = > ? = E

1 E >' 5= > ? ' ' ' -+/ +/ 0 ( 5 5 ' -(/ 0 % ' -(110 % " % ; % % -J30 (

REMARKS

'

" - V =! B > ! C D = *> AI ! , 5 ! & 6 ! , 8G

' " & ! # !

, 8G

( ( -8((340

'

B

4.5.6 Forbidden Subgraphs DEFINITION

'

= > 6 - # 5 ! => 6 => 6

; ! # 5 5 # # = &>

Figure 4.5.3


!" - ;

272

Chapter 4


FACTS

'

-.2H 30 "

½ -

= >

=>

'

U

-8.J110 6 # ,

-

'

-8L/10 "

'

-2H 30 "

'

-8/0 "

'

-( 2/40 "

-

-

-

1

Other Forbidden Pairs B ' T 6 ! -8/0 -( 2/90 1 + !

FACTS

'

,

?

+

+

6

, +

-

, +

?

=

-

>

-

=

-

!

1>

'

-( 2/90 :

, +

,

'

?

+ 9

?

1 6

-2: 0 :

9

?

= > A

, + ? > , + - -

=

=>

,

-8/0 -( 2/90 :

9

'

9

?

!

>

?

;

-( 2/90 "

!

Claw-Free Graphs " !

=

6 ! B

'

9 - - - - ?

6 B '

!

" !

T 6 ! ! -( 2H :10 ! !

!

&

Æ

( -( 2H 0

Æ ! 8& -810 ! ! " -( 2H 0

!


Section 4.5

273

Hamiltonian Graphs

DEFINITIONS

' ( # ( -- =(>0

( -- =(>0 =- =(>> =; ! ! >

' 6 ! 2= > , # (

' 6

5/=

>

FACTS

'

-( 2/90 : , + =, + ? > 1 6 , + # , ? + - -

-

' -/90 "

'

-/90 :

2=

> ! ,

=>

5/=

! 6

= > =>

> ? 5/=2= >>

2=

> ?

REMARKS

'

6 ! @ = -8970> ( -8110

' 8 ( 73

!

2=

>

'

6

! ( -8930 -8930 -+ 2 3 0 -8/40 -2 /70 -2/0 -210

References -8 30 Q 8 2 * # + % , =/3> 4K7 -8 8: L110 . 8 * H 8 : * H L F

$ // =111> 9K -8 8L: 3/0 . 8 * H 8 * H L :

* & F + % , 9 =/3/> F 9K


274

Chapter 4


-8 $L/10 . 8 $ A * H L : ! $ 9/ =/3/M/1> 4/K91 -8970 H 8 L G

=/97> K4

$

4

-8( 2:/90 8 2 H ( H 2 : : & ; + =//9> 74K9 -8/0 8 - . 6 $ N //

-8930 H 8 * % 8 & + A : =/93> -8 :+ 370 F : 8 A J : H + N ;# =/37>

:

DE" !E" ;#

-8((340 8 8G 6 " ( $ ( ; $ % F! 49K9

$ %

-8H /30 H 8 8 H & L B + % , 9 =//3> 74K94 -8990 H 8

+ % , =/99> 31K3

-8930 H 8 * ! % *

F SS" =/93> K3 -8/40 H 8 8 K ( A =//4> 4K1 -8/10 H 8 &

J! =//1>

-810 H 8& ( $ 4 =11> 9K97 -8.J110 8 I ( ( . A JI : = ! > %($ + 1 =111> 77K 799 -8110 * 8 Q 5G [& " ' 7 =111> 9K 3 -8L/10 * H 8 * H L ½¿ $ 8 & A 8" + L $ + Q =//1> 3K /


Section 4.5

275

Hamiltonian Graphs

-8L/0 * H 8 * H L : ! + 4 =//> /K3 -+ 90 2 A + ; # % $ 3 =/9> K 3 -A90 L G AI =/9> K

%

$

-(/ 0 $ ( * + % , 7 =// > 4K7 -(110 $ ( * ) % 7 =111> 7/K 1 -2 /70 H H 2 *

$ 47 =//7> K3 -. 40 2 . =/4> 7/K3

0 $ %

-.2H 30 . .@ H 2 $ H (

2 H . 2 : : & . : & =/3> /9K7 -A/0 4K41

%7 + $

-A!310 A * ! +

% , / =/31> 1K1/ -( 3 0 2 * ( F! Æ , 9 =/3 > K9

+ %

-( : /30 ( H : * / =//3> 4K44

+

-( 2/90 H ( H 2 $ 9 =//9> 4K71 -( 2H 0 H ( H 2 $ H ( ' -( 2H 0 H ( H 2 $ H ( ' Æ -( 340 H ( A 5 .

% =J $ /3 > + F! K / -( 2H :/0 H ( H 2 $ H : : & ;

. % $ 14 =//> 7K9


276

Chapter 4


-( 2H :10 H ( H 2 $ H : : & ! $ / =11> 9K3 -( 2J: 10 H ( H 2 J& : : & "

. + F =11> //K1 -( 2/40 H ( H 2 Q 5G [& " ( ' 1/ =//4> K -(9 0 * ( 6 B ! + % , 7 =/9 > /K -(370 ( ! Æ + 1 =/37> 14K 1/ -2 H9/0 $ 2 . H ( 1 ' ( F! -2: 0 H 2 6 : \ & ( ' -2/0 H 2 N K + 4 =//> K49 -210 H 2

K / F =11> 9K4 -2*//0 H 2 A * ,

( 7 =///> 7K 3 -2H 30 H 2 $ H ( $ =/3> 3/K/7 -2730 A H 2 ! 08 $ F =/73> 4K43 -2$ 970 8 2 H $ & 5 $ =/97> /K/7 -* F+740 ( * H F + ; $ , =/74> 91K91 -*/10 2 * A# $ 34 =//1> 4/K9 -*/0 2 * A# + % , 4 =//> /K -*110 * & : ! $ =111> 94K31


Section 4.5

277

Hamiltonian Graphs

-H 9/0 8 H & A 5 +

0 $ % => / =/9/> K7 -H 310 8 H & * + % , / =/31> 9K 7 -H +3/0 8 H & + + ! B + =/3/> 499K431 -H930 * H ; # , $ =/93> /K -J //0 * J 2 F G &I &! ; + =///> 9K4 -J 470 6 J & ; ) % =:> 7 =347> K 3 -J30 H JG A G : # $ =/3> 44K7 -J /70 H JG 2 F G &I A G ; B ) / =//7> /K -J /30 H JG 2 F G &I A G 5 =//3> K71 -J970 . J AI G * %8 $ 9 =/97> 971K97 -J/90 $ J # +

0 =//9> 4K3 -$ 970 $ $ =/97> 9K 1

#

-$$70 H $ : $ ; ( +

$ =/7> 7K74 -;710 ; ; $ $ 79 =/71> 44 -;70 ; ; * + $

=/7> K9 - 970 : G * $ =/97> 4/K7 -+/0 + F + ) % =//> 9K4 -+/ 0 + F + ) % 4 =// > 7K9


278

Chapter 4


-3/0 $ 5 4 $ 3 =/3/> 41K44 -/90 Q 5G

[& ; ! + % , 91 =//9> 9K -/90 ( 8 * ! + % , 4 =//> 9K/ -/0 * $ /1 =//> 7/K/1 -6930 2 6 * B

$ =/93> 4/K73 -630 6 + 9 =/3> 7/K97 -6/70 6 ; 4 =//7> 9K -6/90 6 + %

, 9 =//9> K -6/30 6 " + % , 9 =//3> 1 K1/ -6 70 + 6 6 ; + 0 $ % =/ 7> /3K1 -6470 + 6 6 $ % 3 =/47> //K7 -+ 2 3 0 . + H 2 K * $ 4 =/3 > /K1 -+930 . + Æ + % , 4 =/93> 3 K37 -Q 970 H Q & 4 +

% , =/97> 7K -Q3/0 $ Q . 3 =/3/> 43K73


Section 4.6

4.6

279

Traveling Salesman Problems

TRAVELING SALESMAN PROBLEMS

& +' $

7 6 6 7 A# 7 * 7 " * 74 6 2 6 77 6 L

Introduction 6 6 =6 > " 6 Æ 6

# - 10 " ! 6 2 6 L

4.6.1 The Traveling Salesman Problem J $ -$0 ! , 6 =6 > * # ! ! & " ! ! ! ! ! =( ! 6 -*+340> " 6 Symmetric and Asymmetric TSP DEFINITIONS

' # 3#& =#3#&>' 2 = > ! ! , = > !

'

3#& =3#&>'

2

!


! ! ,

280

Chapter 4


' 6 / 3#& 6 ! A ! A !

'

6

#

8 3#& ! 6 6 Matrix Representation of TSP

A 6 ! # ! # 6

6 DEFINITIONS

' 6 = > 6 # % ? - 0 ! ! ! 6

6 # % ? - 0 ! !

' 6 E EXAMPLES

'

6 ! 1 % ? 9

# 7 4 1 1 9 4

1 / 3 1

! ( 7 6 V ? 7 ! / 9 1 9 6 !

Figure 4.6.1

'

,7

6 ! # 1 1 9 1 1 / % ? 9 / 1 9 7 / 4 1


9 7 4 / 1 1 7 7 1

Section 4.6

281


! ( 7 4 V ? 6 4 !

Figure 4.6.2

,,7

Algorithmic Complexity FACTS

' 6 # 6 ! !' ! 1 U ! 6

'

( 6 F B

8 ! 1 ! 5 ( ! !

'

- 2970 ( 5 ?F

! ! 5

Exact and Approximate Algorithms DEFINITIONS

' '

!

= > &

: ! # 6 = 6 == > == > == > ! ! =

' 6 * -Q30 ! #! =!> = == > == >>= == > == >> & 6 = ! == > ? == > FACT

'

-* J10 6 #

#! =!> 6 ! #! =!> © 2004 by CRC Press LLC

! 6 !

282

Chapter 4


The Euclidean TSP

. ( 4 ! ! A 6 ! 6 6 ! , -/30 //7 = ( 7> $ -$ //0 ! = -10> FACTS

' - 992 2H970 A 6 F 6 $ 1 !" A 6 , E 6

' -10 (

' ! =6>> - /30

!" #

1= E

6

!" ! 6 -10

' -6/90 6 # 5 $ A 6 1= > A , 5 F REMARKS

' % =( 7> A

*! ( 3 ' A# B

= # >

# ! ( # !

' 6 ! ' 4 * * 4 $ ! 6 -2340 -H2$ -8 /30 ( 6 ! # -: 10


Section 4.6

283


FACT

' 6 - # # 6 @

=

>V @

°

Integer Programming Approaches

L 6 6 + , =-8:110> '* ** =- 30> -'' -'- = -8 6340 -( :610 -F 10> 6 ! &! =-+/30> 6 = > 6 . (& H -. (H4 0

' .,

( ?

(

= > 1 !

: ! = > 6 6 # ' > ?

( 5 ( ? ? ( ? ?

( + 1 + ¾ ¾ ( ? 1 ?

FACTS

' 6 , #

# # # 6 ! ! 5 # *! " ! # 5 = - >

' 6 - * B

+

+

' 6 ! * ! * 1= > -+/30 ! # 5

! !


°

"

&

? 6

284

Chapter 4


;! ! ( # ! B 6 6

4.6.3 Construction Heuristics #

!

Greedy-Type Algorithms

6 6 ! # #

Algorithm 4.6.1: # 5 => + '? +

EXAMPLE

' + >> 6 A# !

( 7 # >> # # # 6 ! 1 =! ! >


Section 4.6

285


Figure 4.6.3

,7

# -H2$> U ! 111] # 6 -H$10 ! >> ! A 6 >> 6 Insertion Algorithms

, 6 ( 6 !

! # ( 6 ! 6 # !

6 ! 5 B ! 6 DEFINITION

' : ! ? ½ ¾ ½ # B # ! ( = > ! = > = > ! = > = > = ( 7 > 6 ! = > 6 = > ? = ·½> / ! = > ? ½ ¾ ½ = > ? = ½> ! = > ? ½ ¾ ½

Figure 4.6.4

2" /"9 " =

>

REMARK

'

'* , , , ! , ! @ # A ! -^0 #


286

Chapter 4

Algorithm 4.6.3:


1"9 2" 3124

#' # - 0 5' 6 ½ : ! " ! ? ( 7 ? : # ! -^0 " # = > ! ? ! ! = > ! = > = > ! ! '? ! = >

2 # ! = ! >

!

= ! > ?

DEFINITIONS

' '

6 567 # !

6 )67 # 6 = ! > ? = ! >

' 6 ,67 # ! # 6 = ! > ? # = ! >

6 # B

! A 6 = -H$10> # ! # 6 -22 =-H2$ 6 #

" , : J -210 : J 6 6 6 = -H2$ DEFINITIONS

'

( 6 8 ! 5 ! ! = ( 74> ; ! 6 = #>

Figure 4.6.5

!

2

" "$

2

' ( #

= >

' 6 , U @ => => =

> 6


Section 4.6

289


FACT

'

- $ J 10 & # ,

= > !

1= >

6

Exponential Neighborhoods 8

_=

>

# #

= >

'

6

( 6

! # !

=

>

" # 6

!

1= > =1= >>

6

-A; 10 -.+110 -2

4.6.5 The Generalized TSP 6

1' !

# 6 -( 610

DEFINITIONS

'

6

1 ( 3 # & 13#&'

2 !

,

! # = > #

?

'

6

# = >

#

'

6

1 ( # 3 # & 1#3#&

!

REMARK

'

; B W % W# % 26

26 ! B

6 W# %

26 26


290

Chapter 4


Transforming Generalized TSP to TSP ; ! 2 6 6

6 Æ 26 6

26 6 -F8/0 -: //0

FACTS

'

" -F8/0 26 6

+ , 6

# 6

4

! " ! #

'

" -: //0 26 6

! , Æ ! ! ( #

!

4 !

!

4

¼

6 !

!

!

?

!

½ ½ ½ 4 ¼

¼

¼

"

!

26

6 ! ! !

! 26

6 ! !

4

=

4

> !

! 8 #

¼

!

26

'

( -: //0 -F8/0 5 !

*

&

#

Exact Algorithms FACTS

'

# =-82 !

-F8/0 -: //0 2 6 *! B 6 % = > ! =>

'

26

-( 610 : 26 -F8/0 6 # 5 &

6 26

'

% ? = > ½ ¾ % #

# ½ ¾ F ? = # B > -2 :

=

>

B

8

8

8

! ½ ¾

# #

# 26 -82 6

!

! >

=> ( !

!

=" -( 610

-82 :

8

B

*! -82 6 L L ) 6 B

"

DEFINITIONS

'

2 !

1

! ! ! " ¾ ?

# 1


"

!

1

65&

! !

?

292

Chapter 4

'

6


6 5 & 65&'

E

2 !

1 ?

! " , L ! !

REMARKS

'

; '

! = #> !

* &'&

! @

@ - J + //0

'

" L =

½ >

F

= W %> L -L /70

Exact Algorithms FACTS

'

6 Æ # L

=-8*110 -F 10 - J 60>

'

( L #

-6L 16L 10

'

L 6

3 ! " -

L ! -8 10

6 # ! L

6 ! 11

# ! L ! ! 4 - J 66L 10 ; L

& #

6

L

Heuristics for CVRP L ! '

L

B &U

6 &

@ ! B

=-A;0 -2: 10 -6 /0 -6L /30>

( L

B & )# # ! L '

&-

+

REMARK

'

6 ! L

L ! W % W % W % W%


Section 4.6

293


Savings Heuristics

6 & + -+7 0 &! L * ! ! -2 /0 -+7 0 -: 10

= > ( # + &=+ > = # > !

6 + => 6 # + =+ > ?

¾

DEFINITIONS

' !½ !¾ =!½ !¾> ! # B =!½> =!¾> 6 !½ !¾ # " = = =!½> =!¾>> ">

' 2 !½ !¾ =!½ !¾> 7=!½ !¾> 7=!½ !¾> ? &= =!½>> E &= =!¾>> &= =!½> =!¾>>

°

'

: , ? !½ !¾ ! / ! ! # 1 6 %=,> ! /

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310

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Section 4.7

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315


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318

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Section 4.7

319


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Section 4.7

321


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Section 4.7

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324

Chapter 4


References - 790 $ ; $ G 1 =/79> 47K7 -&J390 8 & 8 J ; (*** 7 =/39> 334K333 -/40 F 6 ! +

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7 ! $ 9 =//9> /K9 -8 ( ( /90 $ 8 H (P

$ ( A# ! $ 79M73 =//9> 34K11 -8 ( /70 $ 8 $ ( .

+ =//7> 3K/ -8*/40 H 8 ( .( * . !&' + =//4> K1 -8; 10 :+ 8 & ; ; A 6 $ 4 =11> K 4 -8 / 0 F 8 N // -89/0 8 8G ; ! B $ 3 =/9/> K -8(910 ( 8 ( '. =/91> 7K3


Section 4.7


325

-8* J 30 ( 8 ( * H J 2 !& ' '. =/3> 49K7 [ QG +

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6 H 8 L : //4 -. //0 $ . = > $ // =///> 19K4 -. L/40 . & : L& F! Æ B 1 =//4> 91K93 -. L/90 . & : L& . B # + 7 =//9> 9K -. L110 . & : L& . B # B $ =111> 9K -A340 * A :! + / =/34> 41K4 -( ( 3/0 H (P $ ( $ # + =/3/> 749K773


326

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:GG G G $ 0 38&6 * E 8 + * + 5 2 .* E 4 9 1 + 0+ S S+ 5& 38 6 8H + /+

+! B .50+ $I5

38 *B&6 * 8 ' *+ ; ' , ' + $ 7 .5B&0+ $&IB7


Section 5.3

401

Independent Sets and Cliques

3D176 F W D F # 1+ 1

: Æ ' + / .770+ I7 3D B6 D 22+ ; , 2 2 ' ' ' ' + 5 .5B0 BI B5 3D 1B6 D 22 1 + ; ) '! ' ' + & .5B0+ 5I$$ 3E@&56 4 E9 4 ; @ + !F + 5&5

7 +

3E E $ 6 4 9 E

D E + % + IB

.5$ 0+

3E @ $ 6 1 @ E

/2 @ + E ' ,

' '

' 1N+ & .5$ 0+ I 7 31&6 D D 1

+ S ' - 2 ! + * $ .5&0+ BI$ 31& 6 D D 1

+ 2: @ + * B .5& 0+ IB M 3%76 @ 2 S * 4 % _ + K + 7+ 2 += ! 3 " = , 1B+ 77+ 1 ' - ' ( M 3%76 S * % _ + ; , ' !, + & B .770+ I$ M 3%76 S * % _ + ; ' ' + 7 .770+ 5&I7&

3SS576 S E S ; 1 S + ; / ' -

+ .5570+ 75I$ 3S 56 E S @ F + E/ ' ) , ' + ,7 7 !B .5 50+ $I$& 3S45 6 E S ; 4

+ # ' ! ' + & &'! .55 0 I7 3* B$6 4 E * + ; ' + .5B$0+ I7

&

3* @76 # * @ @ + ; !! ' ! + / , + B .770+ $I& 3@6 " @ *

+ E , - + S + 77


402

Chapter 5


3@1 W76 @ @2+ E 1 2+ 8 W/2+ ;

'

, + $ .770+ I 3@56 @ E @2 ,+ ; , ' + .550+ $I$ 3@576 T @ + "

+ / .S E S 4 " * + 0+ / , .5570+ $I$B 3@ 5$6 S @ E + 1 ' + 34 15$6+ I+ 55$ 311&&6 *

1? ;

.5&&0+

1 ? ,2+ # + &I $

31 11BB6 1 + ; 12 F 12+ 1 , ! ' ) + , 372,28 5BB 31(;-@&&6 @ 12+ E (+ F ;- ( @2,+ ; , ' + $ .5&&0+ 7 I & 39B6 8 9+ ; , ' + " D 1 E + B!&!5+ 5B 39$&6 F @ 9'+ 1 - ' + .5$&0+ 7I

+

39B$6 F @ 9'+ 1 ' ' + ) 7 .5B$0+ I&


Section 5.4

5.4

403

Factors and Factorization

FACTORS AND FACTORIZATION !" # $

S !# # # # / *'

Introduction 1 - ' , 2 ' ' / ,

' (+ ' / ) - ' 3 .' Ü 0+ ' ! ? E -+ $ .' Ü 0 -, ' !' + , , ' ' : - + - , ) ' ' ' '

5.4.1 Preliminaries DEFINITIONS

: - . + 0 + , " ' ' " ' : ; ' !

: ; ' ) ' ' -

: ; ' ' !

-

: (' !? ' ' ½ + , ' ' FACTS

1 , 2 )

' ' / , ! ' 4 S 1 ' "M ' '

)'

: 3SB56 ; ! , !' . !' 0


404

Chapter 5


: 3SB56 - ! !' .+ + ' / !' 0

:

3"M B6 - !! . C 0! !'

REMARKS

: 1 ' , ' ' ;2 8 3;28B 6

:

1 , - # + ) !' ) ' ' ,

: ; - ' S K ' / ' ' 3 5 6

5.4.2 1-Factors 1 ' ' , 9 - Ü ( + , ' !' ' ' Conditions for a Graph to Have a 1-Factor DEFINITIONS

:

; . 0 ' ' - !? , .0 : 1 ½ ¿ ' " ; ½ ¿

"

: - ' - ' - - ? : 1 ' +

.0+ ) C ,

,

6 #. 6 0 6 .0 , 2 - 6 .0 #. 6 0

' ' 6

:

1 ' B . 0

- ' ' ? - 0

'

:

1 '

:

; '

+

.0+ / '

' - -

: 1 ' - '


+

G.0+ / '

432

Chapter 5


: ; + $ + ' -

!?

: 1 ' +

.0+ / '

: ; ' ' - - ' 1 / ' ! - '

.0

: 1 ' +

.0+ ' - ' @ . Ü 0 + / ' ! -

: 1 . 0 . < . . 0 ' < . 0 - ): . C0 . ' ' . C0

FACTS

: @

. + ' ,

? -

.0 < .. 0 .0 < ..0

:

#

.)0

+ .0 < G.. 0

G.0 < .. 0

.))0

: ( ! - ' - ' + ' L @+ - L 1+

.0 .0 .0 G.0

.) ) )0

EXAMPLE

: 1 ' # ' / + + 1 . ' / + + 3 1 ' / + + 3 1 . ' / + + !? 1 ! - .2 #0+ .3 0+ .! 0 ' + - .2 #0+ .3 0+ .! 0 '

. b

a

c

a f

f

e

c

e

d

d c

G b

Figure 5.5.1


G

#$ * & -#- .,

Section 5.5

433

Perfect Graphs

5.5.2 Graph Perfection " + " 3"$6 ) 3 ' .0 < G.0 ' ' - . ! 0 - ! " ' + ." 0 < G." 0 " , .f0 .ff0 -+ " , ! ' @ "K ' ' ' 3"5&6 ; , ? ! ' ! ' - + " ? ' ' @ S' ? 1 - , , , ' ' + ' .fff0 - - .f0 .ff0 I + + - , I DEFINITIONS

:

; " < . ¼ ¼0 ' < . 0 ' , - C ¼ + . C0 ¼ ' ' . C0

: ; ' ' - . ! 0 - ! " ' . " < 0+ ." 0 < G." 0 : ; ' ' - . ! 0 - ! " ' . " < 0+ ." 0 < ." 0 : :

;

' ! ' ! '

; ' , ! - -

: :

, + +

; ' ; / ,

FACTS

: #$ $-: .D -N/ 3D &60 ; ! ' ' ' ! '

: 1 S' 1 : ; ! ' ' ' ! ' .F+ , ' ' - + , ! ' ! ' 0

: .D -N/0 3D &60 ; ' ' ' ." 0." 0 ." 0 ' - . ! 0 - ! " '

: $ ! #$ $-:

. -2+ * + @ + 1 3 * @1760 ; ' ' ' "


434

Chapter 5


EXAMPLE

:

- ' # < .½ 0+ -

.0 < .0 < ! + - - - @

+ - /

5.5.3 Motivating Applications ( + , , ' 1 ) + + ,

- ' "K

) ' ' 1 ' 1 , - ' , .0+ .0+ .0 G.0+ Æ + - DEFINITIONS

: 1 . 0

- ' L - , ,

- ' ' ,

-

: 1 " ' , " ) ' , # - ' - C ' " + - C " + , C ¼C¼ " ' ' ? ¼ C < C¼

C ? C¼ " < ¼ : 1 " ' ' , '

: .@ 3@ $60 ; S ' ' + , - .. 00 < .. 00

: .@ 3@ $60 1

' ) .. 00

EXAMPLE

:

& 4 4" 1 ! ! ' , +

' ' - . 0 ' 1 / ' ! '

+ , ' / .. 00

+ ' -

+ 2 ,

, 2

!

# + )- . 0 ! + ' , .. 00 < ; ' ! - ' + +

( ! + ' 1 , . 0+ !


Section 5.5

435

Perfect Graphs

' L ! ' ! .. 00 %- ' ! , , ! +

' , ' ' L ! # + - ' ! 1 ' . . 00½ + ' .. 00 ! ' ' + '

' . . 00½

' .. 00 1+ , " ' + .. 00½ REMARKS

: D -N/ 3D &56 - @ ' ! - ½ < ( . 0 : 1 .. 00

. . 00

/ ' ! (' S ' + .. 00 < .. 00+ @ .. 00

: % ,

.. 00 . . 00 G. . 00 G.. 00 ' , ' , ' . 0 ' + . . 00 < .. 00

'

: 1 / , !

- 2

' L

1

' @ 3@76 ' EXAMPLE

6 - .- 0 ,2+ , $ % 2 ' ! 2 ' ,2 - ' # + ' ,2

- , 1 Æ

- + ) , ! ,2 ! '

, # )- , 2 - L P 1

' : " 4 " 1 '

B

Tour B Tour C Tour A

Tour E

A

E C D

Tour D

Figure 5.5.2 7 & 0 & * $ &&* #$,


436

Chapter 5


- ' + ' , - ' 1, - ? ' , - 9 + L , 2+ - . 0 '

# )+ !

- ,

) ' ' ! 1 ' ' ' Æ + ' , ' + + ) ' , 2 2 '

" ,! - @ S' ? +

2 '

5.5.4 Matrix Representation of Graph Perfection ' ' /!

1 ' ' , , +

7 .

0 '

!

1 .

0 '

9 ' +

7 .

0+ , '

'

@+

1 .

0

DEFINITIONS

:

1

- ' '

'

:

! -

; 7!

7

!- -

8

1

.

0 '

A + .

7! , , 0 '

!

' ' - !- -

+

- - ' :

78 1 E

:

7

8

?

.

0 '

, , - ' '

' '

! -

7!

A + .

0

!

FACTS

7 1 . :

# ' + ' ,

.

0
88GD 8B:A ;* -::-= = ) = = < ;+ & = > -9B: 8B:A 88GD ;* -::-= = ) = = < ;+ & = > -9B: ;* -::-= = ) = = < ;+ & = > -9B:

9 &

( ( ) %

9 ( %

9 $ (

9 )

& EXAMPLES

2 ) ¼ ( C (

) E ( % 5 % (

%

2 3 ) ) ( ) $ % (

& = ( % (

& $ (

+ ) , 2 3 + , 4 ( + , +, , +, H , +, B, +, H B, , ( ) 7

: 5+ , 4

2 % + % ,

¼

2

> + , & : &( +

, : &( +

, & & : &

2 ) +(

&, & ( ) & Strips


Section 6.1

499

Automorphisms

DEFINITIONS

2

%+ ,,

+

2

& )

+ , %+ ,

+ ,

%

( ½ + ,(

+ ,

%

REMARK

2

$

(

) % %

½

(

$

9 +5 6K1!!8,

FACTS

2

7 % ( : 2

1- ( 1 1 ( E ) % ( # ! +?"?,( M " 61""8 K > # > 1- ( C %

( +?"",( ?M B 6 "8 $ -( N % ( $% &' +?",( ?MBB L 6= 8 = ( ') 1 (

+?

,( M!

6= "8 = ( N #& C C 1 ( ( +? ",( ?M! 6= !8 = ( % % ( $ " +?!,( M"


502

Chapter 6

ALGEBRAIC GRAPH THEORY

6= "8 7 = > # ( E ( # 5 = +? ",( M ? L 6=! 8 N =C ( ') L N TL ) ( ! $ !+ ( ?!

) *

6= "8 N 7 = ( ) ( +?",( BMB L 6$ ?8 1 $ ( ') & - - ( ! +? ?,( "M

#

6$ !B8 1 $ ( ) ( 2 &! ! +I : # 5 KS A I : A U ( = ? ?,( >U ( ( P 5S ( J = ( ( ?!B( M 6$ !8 1 $ ( C - ( ( / &&( +?!,( BM?

+ ,! - .!!

6$ $C!8 1 $ = $C) - ( ( +?!,( !!M" 6$ G BB8 1 $ 5 G VC( 1 D 5 ( $( J Q -( BBB

!

# !* % " ( K

6$ 5""8 1 $ J 5 ( ( +?""R"?,( M! 6$"!8 P $ ( E ) & ( +?"!,( !M" 6K"8 = K( % ( "M"" 6K?8 = K( E % %& % ( M

)!

! )!

+?",(

)! +??,(

6KJ ?8 = K J ( N % ( ( +??,( "MB 6K1!!8 = K # > 1- ( E % % ( ! +?!!,( M 6K1!!)8 = K # > 1- ( E % & ( )! ( M 6K1"8 = K # > 1- ( 7 % ( ! +?",( ?M 6K1"?8 = K # > 1- ( %

( +?"?,( M !


Section 6.1

503

Automorphisms

6GL 8 N GL ( ( 3 C ( ?

# ( -

P

6J "8 3 J C( E & C ( +? ",( ?M 6J 1!8 3 J C # > 1- ( ) ( $( +?!,( ??MBB" 6J 1!)8 3 J C # > 1- ( ) ( $$( +?!,( BB?MB" 6 1?8 J # > 1- ( E ) ( " +??,( M

65!8 5) ( ( +?!,( M 65 B8 5) ( (

0! +?

65 8 5) ( & ( !M!" 65 8 5) ( P& ( 65 !8 5) ( W (

B,( M!

) - +?

! +?

% ( J

,(

,( M"

( # ? !

65?8 J 5 (

( +??,( M

65?!8 J 5 P $ % ( : ( +??!,( BMB 61"?8 I # > 1- ( $% & ( ( ( +?"?,( "M 6"8 P $ % (

( $% - +?",( J ( BM! 6 8 1 (

# !( ' ( ( ?

61!8 # > 1- ( E ) (

+?!,( ?MB

61!8 # > 1- ( E 1( 2 # &! !( +Q ( N 3 -( 1 ( , 5 P ( A ( ?!( BM 61!8 # > 1- ( (

(

( 1 +?!,( BMB 61! 8 # > 1- ( (

+?! ,( !M


504

61?8 # > 1- ( > ( BM!

Chapter 6


+??,(

61B8 # > 1- K > ( C %

( ( L 61!8 # 1 ( ') L ( ?M

+?!,(

618 = 1 ( I ( +?,( BM "


Section 6.2

6.2

505

Cayley Graphs

CAYLEY GRAPHS I $

5) 7 C 7

Introduction :

: E

I )9

6.2.1 Construction and Recognition 1 % ( % ( ) ) % 1 I % I % ( I % I % ( DEFINITIONS

2 3 ) % 3 ) ) 4 ½ ( ( ½

( I+ < ,( %2 I+ < , < 9 2 < I+ < , < 4 2

1 : $ ) ) ( ) ( 4 < 4 2 H

I

I+ ,

2 I 1 I +< ,

2 2 2 2

) 2 +,(

2

#

+ ,(

4

!

Figure 6.2.1

)

-

- "- .' $ *"" "

FACTS

2

> I &

2

I I+

2

65"8

) / + >

/>

2

/ > < ¾ > 4 / ( > 4 < > 4 H / / 4 > 4 / 4 H < / 4 !( > 4

/

# + ,

/

4 !(

/

# > # + ,(

/

# > # + ,(

/

4 !(

>

/>

/ + >


>

# + ,

>

4 <

) / + > + />

/>( / > ) ( />( / > ) ) 2 ·½ /> 4 + H ,+ H ,( $<

/>

/>

4 + H ,+ ,(

4

H ,)

/>

4 + ,)

4 +> H ,)
,)(

/

< / > /

>

4 +/ H ,)

>

4 + ,)

/

4 / H (

4 +' H ,)+' H ,(

'

/

/

4

'

' H (

/

/ ( >

/ ( >

4

4 +' ,)+' ,(

) +'

4

H (

)

,)+' ,( 4 +' ,)+' ,(

)
4 ( 4 H ( 4

4

4 ?<

/

4 !(

>

4 !

4 !

RESEARCH PROBLEM

42 $ ) ' ( B ' Y - C )

6.2.3 Isomorphism 5 - I :

I


Section 6.2

509

Cayley Graphs

DEFINITIONS AND NOTATIONS

2

I I+ < , " I+ < , 4 I+ < ,( & %+ , 4 +,

2 2

" I

I$

3

2

3 4 /½½ /¾ / ) C 7 , B , + ( + , 4 +, , , ,

,

: $

,

# ,+ / , B ,

+ /

( 4 $

3 & +, 4 ,+, 2 B , + 7 ) & $( & +, & +, , 4 B " 1 ( & & 2 3 4 /½ /¾ / ) C $ 4 + , $

+ , J

4 B( 4 $

3 + , )

( & $ $ ( &( 4 ( ( +&, 4 +&,) $ ( +&, 4 +&,( +&,

&

EXAMPLES

2 I +!< , I +!< , - 2 2 N% ) ( )

2

7

4 ( 6 4

4

? ? B 8

? B ?

I +< , I +< , ) E (

4 % ( I$ FACTS

2 6A!!8 3 ) I % I$ ) + , 9 + ,


510

Chapter 6

2 63 B8 $

I$ (


)

2 6#?!8 I$ 4 ,( :

B (

,

" ? "

2 3 / ) / I$

E (

/ 4

/ H

( I$ :

E I$ I I$ ) S * 6A 8 &

2 6 !8 $ / ( ) & /

;+, /

+/ ,) ; >

2 6 # B8 $

4 / / / ( / 4 ( )

/ / /

;+,

½ ¾

;+,;+ , ;+,

& Ê

- ) ( - ) & $ / 4 ( & 4

& - ) EXAMPLE

2 1 7 4 B ) +(,( +(,( +(,( +(,( +(, +(, ( & ( +(, 5 / 4 ( & 4 ( ) & 4 +, 4 7 & 4 ( +&, 4 & 4 - ) 7 & 4 ( +&, 4 & 4 "( ) ) 7 ( & 4 ( +&, 4 & 4 " ) 1 ) % B RESEARCH PROBLEM

42 7 /(

I$


Section 6.2

511

Cayley Graphs

6.2.4 Subgraphs : ) I 5

& (

$ 9 &

& )

DEFINITIONS

2

½

( (

2

)

(

( 4 ( ! "( ( ! " ! "

%&

)

! "

4

(

=

(

$

- +

2

0 4

)

4(

0

=

FACTS

2

3

$

2

)

(

$

& $

(

&

& (

2

6#!( 1!B8 $

&

2

6 ?8 $

I+

2

2

I

( =

/ ( /

512

Chapter 6


6.2.5 Factorization DEFINITIONS

2 2

( ) ½ )

2 =

( =

2

) ) FACTS

2

65"8 > I 2 4 '<

) <

C

2

65"8 I

I C 2
I 2 $( )! +BB,( !M? 6A!!8 3 A) ( $ )

( " ? +?!!,( ?M

6A?8 3 A) ( ( ( (

- 2

! 3 &&( 3 ( # L 3 3 SC( #$ J = ( ??( !MB 6A 8 J A 9( S * ( I ( 1 ( J Q -( ?


!(

514

Chapter 6


6IZ"8 I I I J Z ( E

) ( I ) # P$$$( ( ! ! ( 5 P ( ?"( M 67 ?B8 K 7 -( E ) ( +??B,( BM 6"!8 K 3 1 -( ?"!

#

" # ( 1 ( J Q -(

6$B8 # $ I > ( E I & ( +BB,( M? 63 ??8 I = 3 ( 7 I$ ) ( ?M

63 B8 I = 3 ( E % I \ ( +BB,( BM

+???,(

)!

63 ? 8 K 3 ( =

I ) ( ( +?? ,( !M" 63 B8 K 3 ( =

I ) ( ( 63?8 3) C-( I 2 ( & -( ( +??,( M"? 6#!8 1 #( > > ( +?!,( M S * 9 ( 6#?!8 # #C-( E S +??!,( ?!MB

!

)!

V 6 > B8 A ( K 5 S V( K9( - # 1- ( I ( 6 !8 (

( 5 P ( J Q -( ?!

65"8 5) ( E %& ( +?",( "BBM"B 65"8 5 ( E C) I ( +?",( ?"MB!

6 !8 K (

) ( +? !,( M 61!B8 # 1- ( I ( M? 61B8 1 ( # ! 2 # ! = ( ( BB


2!(

+?!B,(

# 5 ( J

Section 6.2

515

Cayley Graphs

61 "8 N 1 ( I

(

6[B8 K [(

+?",( B!M

" !! 2 & (4 -!

( G (

N ( BB

6OB8 [ O( I )2 ( B


)!

+BB,( !M

516

6.3

Chapter 6


ENUMERATION

! I I I I I

5 # N I

I 5

Introduction $ - I ) C :

(

- ) C

$( ( : & ) # : ) ?! - S + 6S "!8 > , 7= 6=8 & (

( (

) I 6I!( I"?8( % ./ #

) A =C 6A =( A =)8

) S 6S "!8( E 6E"8( = 6=?8 & ( ) )

( ) 6=!8 ( - % :( : - % 65 ?8

6.3.1 Counting Simple Graphs and Multigraphs DEFINITIONS

2 ) ( ½ ¾ ( ) )

)

2 4


Section 6.3

517

Enumeration

2

. ) .

% ) 2 ' . +, . +, .

2

. .

)

FACTS

2

) ) , )

5 ) Æ ,

2

+ , 7 , ( ¾ ( ) ) , ) ) ,

2

) )

Table 6.3.1 ,

B

5 )

5$*" * - 2 " "

B B B B B B

(B

B

! " ? B

+¾ ,

"

,

!

"

B ( (BB (BB ( ( (BB (BB ( B

B (B (?" B(? ( ("B B(?B ?(?B (! (! ?(?B B(?B ("B

" !" (! B(! ?"("B ! (!B ("(BB (B"(B (?B (?BB ((B (!("B B((! !(( B B( ( BB

(! "

(B?!(

2 6 8 ) )

4

+¾ ,

4

+¾ ,

"((

(

)

5 )

2 S * ( ( +* , 4 * )


518

Chapter 6

Table 6.3.2


1 " *$*" * - 2

"

!"

! (!B

("

(

" ("(?

) - C ' +* ,( ' 4 7 & ( +* , !( * 4 ( * 4 ( * 4 * 4 * 4 * 4 * 4 B

2

5 + ,

( ( + , 4

]

] ' * ]

+ ,+

¾ ¾·½ ,

¾

= - +* , ( + , + ,

( >& + , 2

4 4 4 H ]

H

]

]

]

4 H ? H " H

4 H B H B H 4 H

H B H B H

H B H B H "B H B

H H B H B

2

) )

+¾ ,

2 +, 4

) ,

2

6=( S "!8 2 +, & ) ) ) & + , ) ) H 5 )

2

) )

& + , ) ) ) 5 )


Section 6.3

519

Enumeration

Table 6.3.3 /* -

, B

! " ? B

2 "

, "

? ?

!

"

B ?! " " ?!

B ?"B ( (! (

(B

(

+¾ ,

2 ( ) % 2 )

, +, 4

)

)

, 2 6=( S "!8 , +, &

) ) ) & + , ) ) % H H H H 5 )

9

½ 9

Table 6.3.4 5

* * -

, B ! " ? B


2 " , "

! " B

! ! ! ? ? ? ?!

" " ! B

" ! ( " (!

520

Chapter 6


EXAMPLES

2 7 0 0 ) % ) B 0 )

Figure 6.3.1

/* - 2 " "

2 7

7 & 0

Figure 6.3.2

0"" * *

* * - 2 " "

6.3.2 Counting Digraphs and Tournaments DEFINITIONS

2 ) ( ½ ( ) ) )

2 + , ( ( ( & ) )

2 + , ( ( &

2

. ) . 4 % ) . 2 + , ' +. +, . +,

2

)

. . FACTS

2 ) ) , )

Æ 5 )

2

7 , ( ( ) ) , ) ) + , ,


Section 6.3

521

Enumeration

2 " ,

Table 6.3.5 5$*" *

* " -

,

B

B ?B (B (" (B "(! B !!(B (?!B !(? B "(! (B"(!

B

B ? !? ? !? ? B

! " ? B

(B?

2 ) ) )

5

+¾ ,

2 ) )

)

(

2 5 + ,

( ( + , 4

]

] ' * ]

+¾ ,

= - +* , ( + , + ,

(

>& + , 2

4 4 H

]

]

]

H " H H

]

]

4 H H 4 H

4 H B H B H H B H B H

4 H H B H

H ?B H B

H H H ?B H B H B


522

Chapter 6


2 +, ) )

+, 4

) ,

2

6=( S "!8 +, &

) ) ) & + , ) ) H 5 )

2

) )

& + , ) ) ) 5 )

Table 6.3.6

- 2 " ,

,

B

! " " " !

!? !B! ( (?B ( !B

"

?( B"

! " ? B

2 2

( ) %

´ ½µ

6N8 ) " ) ] " 4 ] ' * ] +* , C ) -(

+* , 4

+ ,* *

*

5 ) !

2

6# "8 3 " +, 4 H H H H H H ) ( 7 +, 4 H H H H H )

" +, +, 4 H " +, 5 ) ! J &


Section 6.3

523

Enumeration

Table 6.3.7

" - 2

5

(""B ?( ?(!(B ?B(!(" (B"(( "

B

! " ? B

(BB" !"( ?((?? ""( (?B (B(?(!

EXAMPLES

2 7 0 ) % ) B 0 )

Figure 6.3.3

*

* " - 2 "

2 7 E

Figure 6.3.4

- 2

6.3.3 Counting Generic Trees DEFINITIONS

2 ) ( ½ ¾ ( ) ) ) )

2 &( ( %


524

Chapter 6


2 + , FACTS

2 1%*% * 6I"?82 )

5 ) "

2

) ) 5 ) "

Table 6.3.8

5$*" "

" *$*" - 2

3)

3)

? !(!! !( ? (B?!( (B (! (BBB(BBB(BBB (?!(( B !(BB"(!B( "" (?"(B"((" !?(!(!!(( ?(?(? (B(?B( ((?(B( B (" (?"B

! " ? B

2

)

(? ("B! ( (!"(? ? BB(BBB(BBB (!(?!( ? (?!( ( (!?( B(?(B! ( ?(?(!(? (? (?(B "(?(! !(B!(?(B!(?!(?

) &

+, 4

½ &

4 H H H H ? H B H

2 6I!8 Æ & +, )

+, 4

½

+ ,

½

% & S + 6S "!8,

+, 4 & 5 ) ?


+ ,

Section 6.3

525

Enumeration

2 ) "

$+, 4

½

" 4 H H H H H H

2 6 * 6E"82 Æ " $+, )

+, 7 ! )

$+, 4 +, +, +,

5 ) ?

2 I : &

+, 4

½

7 4 H H H H H H B H

2 Æ 7 & )

+, 4

½

+ , H

% &

+ , 4

H

&

½

+ ,

2 ) !

) ) % ) )

2 & ) )

2

) & B

REMARKS

2 2

$ (

E

( ) ) ( )


Section 6.3

529

Enumeration

Table 6.3.10 #

" # " # -

! " ? B ! " ? B ! " ? B ! " ? B

2 "

+ ,

+ -,

" ! ? "? B! (" (B! !( ? ?( "(" (?B (?" "B(? ( (BB ( (B? (!(" "( ?( B("(?! ?(BB(" !( ( ! ("?(??( (B(!B("" ((!("B (" (B(" ? ?(??( ("" !((" (B ?B(?"((B ("!("((? (BB(B( B!( (" (B(B!( ? (B("(B(!" ?"("B(!!?(!B(" ("((BBB(!! !B("B!(?! ("(! (?(BB(BB( ("?

? " ! ? "B ("" (! B(? ("? B( "(" (? ?B(! (!"( " (!("B (?B( (!?!("" ?("?( B(("B !(B( (?B(B!( ((" (! B( B(B!(!? !(!((! ? !((B""( B ""( ( (? ?(!"(?(?B (?(?!(""(" (B(?B(!"B( "(? (!!(!(? ( !(!"(?(? ? ("("B(!(

2 A E )


530

Chapter 6


2 3 : & ( ) ) H ( ( ( (

FACTS

2 % )

4

4

4

H H

H

+,] 4 H + H ,] ]

B

5 )

1* 8$

Table 6.3.11

I J )

I J )

? (B (" (!? "(!" B"(B !(?BB ( !(B ?( ?(" (!( !B

! " ? B ! " ? B

?( (!?B !!( "(!BB (! !( (?B ( (B(B ( ( !(BB ?("( ( B (B?( ( B ("?(?B(!( (" (? (B( "( !((B!( ?((B(? (BB (!!(?(!B( B (BB(( ( ( " ("(?" (B(B?(B (( (B?( (?B? ((B ("!!(B"(?"

! " ? B

2 ) I ) 5 )

2 ) ) 5 ) 2 ) H 5 ) EXAMPLES

2 7 ! ( 7 " ) ( 7 ? !


Section 6.3

531

Enumeration

Figure 6.3.7

Figure 6.3.8

Figure 6.3.9

"" - 2

$% - 2

*# - 2

References 6A =8 I # A = =C( ) ( +?,( BMB 6A =)8 I # A = =C( ) )

( +?,( B!!MB" 6I!8 I ( E ( +"!,( ?MB 6I"?8 I ( (

6 +""?,( !

6N8 3 N ( 5

(

MB

! "

M!"

!

+?,(

6=8 7 =( ) ( ( ( ( ! !" +?,( M 6= ?8 7 =(

# ( 1 ( ?

6=!8 7 = > # ( #


?

(

( ?!

532

Chapter 6


6=?8 7 = ( ) ) ( ( B +??,( M 6 8 > J )( > ) ( 6# "8 K 1 # (

! !( = ( 1 ( ?

6E"8 E( ) (

"

2 ? +?",( "M??

6S "!8 S I ( !( 5 P ( ?"! 6 "8 K (

" +? ,( BM

2 # !* # !*

& !!( 1 ( ?"

65 ?8 J K 5 5 0(

( ??


2 &" 1 !(

Section 6.4

6.4

533

Graphs and Vector Spaces

GRAPHS AND VECTOR SPACES "!# ! $ A I N% I 5) ' I 5) ' ) I I 5) I I 5 N I RIA ! C I I 5

Introduction > )

) ) 2 G 0* ( G 0* E * > ) ( ( ) ) G 0* : ) C ( G 0* : ) C ( % ) ) ( $ ( ( 9

5

) % ) ) ) & ) T # )

C

6.4.1 Basic Concepts and Definitions

( ( )

) I 7 - ( ) ) % 7 ( 6Q??8 65?8

' %(

(

4 ½ ¾ (


,

(

4 + , + , 4 ½ ¾

534

Chapter 6


$ + , ( ) ( ) + , DEFINITION

2 &

) ) ^ REMARK

2

$ + ( ,

EXAMPLE

2

>& ? 7

v1 e1

e3

v2

v3 e2

e4

e5

e7 e6

v5

e8

v4

Figure 6.4.1 Subgraphs and Complements DEFINITIONS

2

4 + , 4 + ,

2

> ) % : ) 4 + , 4 + ,( ) % J )

2

> ) % : ) 4 + , 4 + ,( ) & J & )

2

) 4 + , 4 + ,( ) , + ,


4 +

Section 6.4

535


EXAMPLES

2 7

4 ½ ( )

7 7 +, 7 4 ( & ) 7 +),

v1 e1

e3

v2

v3

v5

v4

e8

(a) An edge- induced subgraph of the graph G

v1 e1 v2

v4 (b) A vertex-induced subgraph of the graph G

Figure 6.4.2 0 "#"" $ " 23#"" $

2 )

7 +),

7 +,

v1

v1 e3

v2

e1 v3

v2

v3

e2 e4

e5

e7 e6

v5

v4

v5

(a) Subgraph G'

Figure 6.4.3 0 $


e8

v4

(b) Complement of G' in the graph G

" *

7

536

Chapter 6


Components, Spanning Trees, and Cospanning Trees DEFINITIONS

2

-

( N% B

2 &

5 $

(

% N% !

2 2

)

& ) & )

2 ) $

"

2 / (

/

2 "

"

2 3 ) & , / 8+ , + , ) 8+ , 4 / + , 4 , H /

EXAMPLE

2

" 7 v1 v1

7

e1

e3

v2

v3

v2

v3

e2 e4

v5

v4 (a) A spanning tree T of G

Figure 6.4.4

e5

e7

v5

e6

e8

v4

(b) The cospanning tree with respect to T

0 " "

FACTS

2

& )


Section 6.4

537


2 & ) , H / / ) , H / REMARK

2

' (

Cuts and Cutsets DEFINITIONS

2 I 4 + , 3 ½ ) 9 ) 4 + ( , & ! " & (

2 $ (

( >: (

EXAMPLE

2

7 7 ( ! "( 4 4 ( ( 7 +, 5 ( ! " (

7 +),

v1 v3

e1 e2

v2 e4 e7

v5

v4

e8 (a) A cut of the graph G

v1 v2 v3

e5 e4 e6

v5 v4

e7 (b) A cutset of the graph G

Figure 6.4.5


0 "

538

Chapter 6


The Vector Space of a Graph under Ring Sum of Its Edge Subsets DEFINITIONS

2

5 4 ½ ) ) )

7 & ( ) + B B B B, ) 7

2

+ ' , ( ( ) ) )

2 , 4 + , 4 + , 4 +? ? ? ? ? , ( ? 4 ) ) ( + + ( ) B 4 < B ) 4 < B ) B 4 B< ) 4 B, FACT

2 , ) , ) , ) + , , 7+,( % (

) + ) , ) _+ , REMARKS

2 2

)

$ ) ) , + , ) ) , ) E) + ^, B _+ ,

2 2

$ ( 6 8

) & 65 8( 6I!)8( 6N!8( 65?8( 65"8

6.4.2 The Circuit Subspace in an Undirected Graph DEFINITIONS

2 & I (

2

) ` + , $ ( ` + ,

9 ^,


+

Section 6.4

539


FACTS

2 2

)

9 ( ) ) )

2

) ( ` + ,

2 ` + , ) _+ ,

EXAMPLE

2 7 ( ( 7 v1 e1

v1 e3

v2

e1

e3

v3 v 2

v3

v2

e2

v3

e2 e4

e7

e7

e4

e6

e6 v5

v4 v 5

e8 (a) Circ G1

Figure 6.4.6

e8

v4

(b) Circ G2

v5 (c) G1

G2

- "

REMARKS

2 2

7 ) P) 6P8

( + Ü,

( ( &

Fundamental Circuits and the Dimension of the Circuit Subspace DEFINITION

2 " : (

$

" (

FACTS

2

" ( , H ( "


540

Chapter 6


2 " ( (

"

2

+, H , ` + , )

2 $ ( ) &

2

) ) ` + ,( ( ` + ,

: , H ( + ,

2

) ` + , : + , 4 , H /

/

EXAMPLE

2

7

"

4

½

I I I I

4

4

4

4

$ ) % ( ( ( ( ( 7

6.4.3 The Cutset Subspace in an Undirected Graph % 5 ) ) & DEFINITION

2 9

) 6+ , ^ ) 6+ , FACTS

2

> 9 ( 6+ ,

2 2

6+ , ) _+ ,

< ( 6+ ,


Section 6.4

541


EXAMPLE

2

I 7 ½ 4 !½ " 4 ! " 7 ( 4 ( 4 ( 4 4 4 ( 4 ( 4 # ( ) 4 !0 4 "( 0 4 4 ( 4

4 4 (

4 4 (

4 4

$ ( ) 7 Fundamental Cutsets and the Dimension of the Cutset Subspace DEFINITIONS

2

3 " ) ( ) ) " $ & " ( ! "

) "

$ ) )

2 &

"

(

FACTS

2

( ) "

"

(

2

) " )( ( ) ) "

2

& ) 6+ ,

2 $ ) ( ) & 5

2

) ) 6+ , ( 6+ ,

: ( - 8+ ,

2 ) 6+ , : 8+ , 4 /

/

2 & ) ) 6+ ,


542

Chapter 6


EXAMPLE

2 7 7 ( " 4 ½ A A A

4 (

4 (

4

,(

A 4 $ ) % 4 ) ( ( ( ( ( ( 7 B

6.4.4 Relationship between Circuit and Cutset Subspaces A ) ( 7 7 5 - ) & & Orthogonality of Circuit and Cutset Subspaces DEFINITIONS

2 ) , < ) < ,

,

2

)

+ , C J C ) ) FACTS

2

)

=( )

2 7+, C

2 ) ) ) )

) ) >: (

2

) ) ) )

) ) >: (


Section 6.4

2

543


)

Circ/Cut-Based Decomposition of Graphs and Subgraphs DEFINITION

2

)

) & J C

FACTS

2

$ ) (

2

6I!8 )

)

2

$ ) ( ) + , ) &

2

6I!)( 1 #!8 > ) $ ) : '( EXAMPLES

2

I 7 ! $ ) % ) ) 5 )

) _+ , E ) ½ ( ( ( 2

4 + B B B

B,

4 +B B B

B,

4 +B B B

B ,

4 +B B B

B B ,

4 + B

B B,

4 + B B

B B,

4 +B B

B B ,

$ ) ) & ( 7


544

Chapter 6


7 ( +B B B ,( ) ) ( ) & 2 +B B 4

+

B ,

B B B

4

B B, + B ,

+ B B B B B ,

( + B ,

e4

e7 e3

e2

e5

e6

e1 Figure 6.4.7 9

**

2 I

7 " $ ( ( ( = ) ) ) & = ( 7 ( ) % 2

+ ,

4

+ B B

B, +B B

+ B B B , ( ( ( (

B

( +B B B ,

v1 e1 e6 v2 Figure 6.4.8 9


e4

e2 v4 e5 e3

,

v3

**

Section 6.4

545


6.4.5 The Circuit and Cutset Spaces in a Directed Graph $ ( A(

&( 0

:

$ (

)

&

( (

)

( ( ( (

Circuit and Cut Vectors and Matrices DEFINITIONS

2

) ( -

- + ,

2

3 ) 4 + ,

(

% )

2

+ , ) (

2

( 4 + ,

% )

2

3 ) 4

½ ¾ (

)

,

+½ ¾ ,(

4

2

B

3 ) 4

½ ¾ (

)

,

+½ ¾ ,(

4

2

B

3 ) 4

½ ¾

3 ½ ¾

½ ¾ ) ( (


$

,

&

546

Chapter 6


,

&

The Fundamental Circuit, Fundamental Cutset, and Incidence Matrices

J&( % & REMARK

2

% % )( ( ( & + ( ,( & ( & ) +½ ,(

4 B

+ ,

+ ,

DEFINITIONS

2

) " ( ) ( ( +, H , ) & & 5

( & " ( ) 1 ( + , ) & & 3

"

2 ( 0 ( ) & & ) & & ) 0

2 & )

: ) & & : ( B EXAMPLES

2 I 7 ?+, ) 7 ?+), +,( + B B B , + B B B B ,(

2 I " 7 ?+, ( ( ( " &

& 2 7 I # &2 I I I


B

B

B

B B B B B B B B

Section 6.4

547


v5 e4

e7 v1 e3

v3

v2

e1

v1

v4 e2

e2

e2

v3

e6

e5

v4

v2 e5

e5

v5

e6 v3 v1

e3

v4

v2

e1 (a) A directed graph.

(b) A circuit with orientation

(c) A cut with orientation

Figure 6.4.9 0 "" 7 7 " - 7

I # &2 A A A A

$ # &2 J J J J J

½

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B

B B

B

Orthogonality and the Matrix Tree Theorem

)

)

FACTS

2

)

'

$

( )

'(

( C

( + ,

3

)

) +,

,

)

% ) ` + +),

% )

6+ ,


)

548

Chapter 6

2

$

2


% $ ( ) &

) Æ

)

) &

5

(

)

Æ )

2

` + , 6+ , $ (

(

2

` + , 6+ ,

%

) ) ( : ,

H /(

( :

/(

- ( / )

2

)

2

2

I "

½

)

)

4

& 4

& C ,

5

4

H 4

C

1

1

4 6@

) & 1

2

1

@

4 64

( 8( @

) & 4

) & 1

8(

" 5

(

@

&

" # ( 1

4

4

&

) 5

( &

2

+

( ! !

& 0 0

, 7 (

: )

EXAMPLE

2

>&

7

REMARKS

2

A

)

% (

2 +,

- H ,(

+ N% ,


& +

% -

,

Section 6.4


549

2

& 0 0 ( $ ) % + , & : & 0 + * , : ) & & + , 7 ) 65?8 &

65"8 Minty’s Painting Theorem

+ , ) DEFINITIONS

2

2

2

&( ( 4 1 C ( ) ( ( ) ( & ) - FACTS

2

+ ! , 6# 8 3 ) 7 ( & 2 & - ) ) ( - & - ) ( -

2

> ( ) ) )

REMARK

2

# * + - , )

+ 65?8, 7 7 E - ) # ) % 6# B( # 8 5

) ) 6PI"B( I! ( 1 !B8

6.4.6 Two Circ/Cut-Based Tripartitions of a Graph $ Ü A ) 9


550

Chapter 6


1 ) ) Bicycle-Based Tripartition DEFINITION

2 ) ) ( ) EXAMPLE

2

½ ( ( ( 7 " )

FACT

2

6 !"8

2

)

)

)

% ) 7

REMARK

2

6 !"8 7 ) 6?8

A Tripartition Based on Maximally Distant Spanning Trees DEFINITIONS

2 ( +" ",( ) % +" ", 4 +" , +" , 4 +" , +" ,

"

2 " " +" ", " "

+

"

" "

,

& ) )

2 & " " 5

" " ) 2

3 A ) " % ) 3 A ) " % ) A


Section 6.4

551


) ( ) : A½ ( A A 4 A ) A + & -

2

) N%

)

2

+

& " " , ')

) ')

) FACTS

2

6G G ?8 3 " " & " " % )

" "

) " " % )

) " "

2

6G G ?8 I 4 + , 3 ) ( ( 4 + , + , ( ( + , + , &

EXAMPLE

2 $ ) % " 4 " 4

& 7 B 2 4 ( 4 ( 4

e1

v2

v3

e5 v1

e8 e4

e2

e6 e7

v4

e3

v5

Figure 6.4.10 REMARKS

2 $ ) + - , : , H ) ) ) )

, H

) : ) :


552

Chapter 6


) ) ) C : ) ) , H ( - E- ( $ C- ( 1) 6E$1!B8 ) ( & ) ( (

) ) : ) ) )

- ) G G9 6G G ?8 ) E- ( $ C- ( 1) 6E$1!B8 ) - 5 65"8 & )

2 3 63 ! 8 A 1 ) 6A1!8 &

6.4.7 Realization of Circuit and Cutset Spaces $ ) ( & -

&

& ) ) 6?8 I) 6I"8 6 "8 ) ) 1 Æ C) & & 5 65 8

) ( & Æ * C) DEFINITIONS

2

7

& - +

v5

v6

v4

v1 v2

v3 (a)

Figure 6.4.11

2

v5

v1

v4 v2

v3 (b)

- & - +

& 7 7 4 67 @ 8( @ &( )


Section 6.4

553


7

2 &

) Æ ( ( B 2

9

(

( B(

& ) + Æ ( ( B, C + , ) C

2

)

& ) * * -

FACTS

2

7 (

&

2 2

0

"

&

7

& ( ( B ( * ) * - ) + - & ,

7

2 & & (

B 4 B B

B

7

B

2

7

2

7

& C) & & G - 7

7

& C) &

& G - 7

7

REMARKS

2

# 6#!B8 * C) ( * ( !

2 > - % 6 B( 8 A 6A!B8 % & ) & & ) &

& C ) A * 65"8

2

& C )

- $ & I)

6I"( I?8 C ) $ ( A 6A!B8 A Q 6A Q 8 C


554

Chapter 6


& & N A * ) ) 65"8 Whitney and Kuratowski

1 ) - ) 1 618 G - 6GB8 1 ) ) ( 1 ) = % ) + 65?8, - ) N% : DEFINITION

2 ¾ ½ ) ¾ ¾ ½ ½ FACTS

2

$ ) ¾

½( ½

2 2

¾

618

$ -( G - 6GB8 ) G - REMARK

2

5 61B8 G - * $ : G - * C) & C )

References 6A!B8 P P A ( 5 2 (4 - &! !( N ( N > > ( $ $ ( #( $ ( ?!B 6A Q 8 7 A N I Q ( 5 -( ! +? ,( MB

&

6A1!8 K A 3 1 )(

( " &! ! +!,( !M 6I"8 $ I) ( # )

( B( ! +",( M 6I?8 $ I) ( & ) - ( +??,( !M!


&% !

Section 6.4

555


6I!8 1 G I( E ( B +?!,( M? 6I!)8 1 G I(

# ( J =

&

( ( ?!

6I! 8 3 E I N # (

-( & ! ! !! +?! ,( ?M 6N!8 J N ( # = ( ?!

4 ! " " (

6> 8 3 > ( 5 ) 5 ( " +! ,( "MB 6 "8 3 ( ( 6Q??8 K 3 K Q ( ???

! " +?",( ?M

# &! !( II (

6G G ?8 G Q G9 ( #& ( & ! +? ?,( MB 6GB8 I G - ( 5 ) I ) ( 7 +?B,( !M" 63 ! 8 # 3 ( ( & ! !! +?! ,( M" 6#!B8 1 #( * C) ( ! +?!B,( B M 6# B8 K # ( # -(

&

& !

% * !+?

B,( ?M

6# 8 K # ( 5 - a * ( &% ! " +? ,( ??MB 6# 8 K # ( E & ( - -

( +? ,( "MB 6E$1!B8 E- ( Q $ C- ( = 1)(

& -( & ! ! +?!B,( ?M?? 6?8 G ( ! ( J N ( $ ( ??

# ( #=

6 8 # A ( 2 ) ( I" +? ,( !M

) I

& !

6 !"8 I ( E ( ! 2 )! ! +?!",( ?M


556

Chapter 6

65 8 5 5 # A ( 1 ( ?


# ! (4 -!(

65"8 # J 5 5 G ( 1 +$ ,( ?"

# !* (4 -! " !(

65?8 G # J 5 5 ( 1 +$ ,( ?? 6?8 1 ( # ( !M

# !8 " !(

! 2

?B +??,(

6 B8 1 (

) ( +? B,( ?BM?! 6 8 1 ( 7 (

+? ,( B"M!

6PI"B8 K P 3 E I( ) ( & ! ! !! ! +?"B,( " M" 6P8 E P) (

!! !( # 5 ( ?

61B8 N A 1(

& # ( =

618 = 1 ( (

( BB

7 +?,( !M"

61 #!8 1 1 3 # #& (

) ( & B +?!,( "M"? 61 !B8 N = 1 ( -( & ! ! !! ! +?!B,( M!


Section 6.5

6.5

557

Spectral Graph Theory

SPECTRAL GRAPH THEORY

%& & A # & 1 - 5

3 ( 5 ( > A N 5 I C 3

Introduction 5 + , + ) ,

A

& ) B 9 & )9 ) 3 I C ' 5 C 6I 5 !8 ?! 5 )9 ( )

6.5.1 Basic Matrix Properties # & - & ) 6# # 8 6 B8 DEFINITIONS

2 ) + , : & 0 + 0 , 0 4

! B

9

2

9 &

2 2

+C

0,

6 )

2 7 (


558

Chapter 6


2 6 0 4 6 FACTS

2 ) 9 &

( . ( ( : & 9 + =

& ,

2 2 2

) :

9 & 0 ) C( ( :

& @ + , @ 0@ 4 @ 0@ ½ &

2 9 & ( ( ( B( 9 & B

2 ( 9 ) - 9 &

2 $ ( 6 6 +5 9 & ) ( - 7 ) M 6# # 8,

2 $ ( $ + : 7 ) , 5 9 &

( )

: ) )

2 3 ) )

2

1 ) )

) +

7 ?,

2 6

)

B( (

6

REMARK

2

A 7 (

) ) (

EXAMPLES

& + (

- ,

2 2

+ ,½ ½ * ½ ) 2 , B


2

*,

Section 6.5

2 2

559


2

+'#) H ,½ ' 4

2

-

2 +'#),½ ' 4 J (

(

2 I+, 5 2 + ,½ B ½

2 2

)

2

1

+ ',+, ' 4 B

+ H , 9 * & 5 2 +'#),½ ' 4 H

2

2

+ + 4 I+,,2 B * * 2 B * *

2 )

1

2

6.5.2 Walks and the Spectrum Walks and the Coefficients of the Characteristic Polynomial DEFINITIONS

2 )

2

& 9 %

FACTS

2

2

$ '

$ 0 9 & ( ' &

2 2 2 2

) -

9 & ( & 0

0

) -

0

) -

0

0

0

'

)

& )

65 8 $ H H H H

( 4 4 B ( ) ) + &

+C 0,,


560

Chapter 6

2


65 8 $ H ½ ½ H H ½ H

( 4 +, ( ) % ' ( +4 , ) 4 ( +4 , ) 4 REMARK

2

7 " Æ! $

6IN 5?8 $ & 7 ! Æ

Æ )

) & ' %& + , 7 ( + , + , (

Æ ) % ' 7 ( ) %( )

+ - ( - , )

Walks and the Minimal Polynomial DEFINITIONS

2

>+, ( >+0 , 4 B

2 * + ) + , H FACTS

2

,+, 4 -

+ 6 ,

2 $ 9 & 0( 0 4 B ' + $( 0 4 B

2

$ , ( , H ) ) $ 7 B

2 2

+ , &

2

) +

7 ( ) \ - \ : ) , EXAMPLES

2

J & ( ( ( I+,( - ( ( 1 )

2

& )


Section 6.5

561


OPEN PROBLEM

4

I C ) : E (

) Regular Graphs DEFINITION

2 ' (

+ 6 , ) ,

2 2

D

2

0

D

9 & : 0 C = 6 ( 6 0 :

6= 8 7 9 &

) / $

! + ,(

+ ,

9 ( (

)

) ) ( ( 0

2

$

,

D + ,(

)

9 ( )(

) ' ( )(

'

EXAMPLES

2

! + , ) 1

$

2

D + , A+ , $


566

Chapter 6


Distance-Regular Graphs and the Hoffman Polynomial

+ / , ( = 0

DEFINITIONS

2

F

) % 0 4 C "

0 4 0 9 &( 0

&(

& + *,

F B

2

F

/

& -

+ *,

FACTS ABOUT THE MATRIX 0"

2

( 2

0

7 ( 0 0

0

4 D (

/ 0

&

) )

4

0

0 0

H + 0

,

0

) ' 4 B

2

) 9 & 0

H

2

= 0

2

)

FACTS ABOUT THE PARAMETERS /

2

- -

4

/ - (

) )

)

2

0 0

- -

0 - ( 0 -

2

-

2

(

Strongly Regular Graphs DEFINITION

2

+ 6 , ) ( ( 6 4 / 4

/

(

EXAMPLES

2

2 A+ , ++

2

3 2 A+


,) ,

, +

,

Section 6.5

567


2

2 - +/ , % % 7+/ ,

9 ( % ( 0 : +

)

/

+/ +/

,) +/

,) +/

) , ,),

FACTS

- &

) & 0

&(

) ( 9 &(

2 +

6

7 + 6 ,( +

, 4

6,

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5 0( 0 ( C (

+ : ( 9( 9 , * C = 0

$

( + 6 ,(

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6,0 H +

,C 4 D

+ 6 ,

2

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6, H +

,

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,¾ 6¾ H ,¿ 6¿ 4

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$ F 7 ! :( ,¾ 4 ,¿ 5

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5 6¾ 4 6¿ ( ,¾ ,¿ :

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2

¾

7 (

½¾ + /

¾,(

H +

,¾ 4 ,¿ 4 +/

6, H + , 4 ¾ H +/ ,)

H ,)(

6¾ 6¿ 4

FURTHER READING 7 ) &

6 B8

6AI J"?8


568

Chapter 6


6.5.5 Spectral Characterization E : - 2 C ) Y 7 )

EXAMPLES

2 7 (

½ B & ) : + - ) , )

5

( : )

)

Figure 6.5.3

** *

2 7

) + ,+ H ,+ H ,

Figure 6.5.4

** " *

2 7 ) )

+ , B

+ H ,+ ,

Figure 6.5.5 1 * - $ 2


Section 6.5

569


2

7 ½ 5 ( ( % $ ( 9 A+ , + , A ( + - " , ) a

b

c

d

a

e

e

f

f

g

g

a

b

c

d

a

0 *% * *

Figure 6.5.6

Eigenvalues and Graph Operations

E ) DEFINITIONS

2 ( ) ( & ) + , & & 9 ( ( 9

2

+ 9 , ( ( ) ( ( ) & 9

FACTS

2

&

2 6

) ) H 6( 6 6

2

3 ) 7 $ 4 B ,( % ! ) - I ) ) ) ) ) $ , $ !

2

65!8 3

+ , )

-#

+ , 4 - ½ +,- ¾ ¾ +, H - ½ ½ +,- ¾ +, - ½

-½ ¾

2

!

½ -¾

¾

65!8 ) ( ))


570

Chapter 6


EXAMPLES % & ) K = 0

+ K = 0 6# !8, 3 ½ )

2

N% !

) - $ ,

( $ 4 B , !

$ -

( (

2

=

! + ,

) )

( )

REMARK

2

) C J>5 ) N I- S

6IN 5?8

6.5.6 The Laplacian 3 9 & ) 9

$ ( (

- )

3 (

- )

DEFINITIONS

2

-

: &

&< 0

$ ( A 4

0

9 B 0(

&

9 &

$ ( 4 C ( 0 A ) )

9 &

3

2

67 !8

6

6

6

3 A

% ) 6

FACTS ) 3 ) 3 ;

4 ; + , ) )

) ) * A

3 A

) &

(

9 A )

9 A

+A ,

2

; + , 4 +

2

9+A, 4 ; D

2

B A )

,

2

6

4 ; + ,


Section 6.5

571


2 5 6¾ 4 6¿ 4 4 6 4 ( ; + , 4 ¾ 2 ) 2

+

6¾ ½

) ¾, 4

6¾+ ½ , 6¾+ ¾ ,

2 6JBB8 $ (

½ + ,

6¾

EXAMPLES

2

6¾ -

2

6¾

2 2

6¾

2

6¾

+

, 4 + +#),,

+

, 4 + +#),,

6¾ 1

+

, 4

+

, 4

+ , 4

,

FURTHER READING

3 ) A # 6# ?8 & ) # J 6JBB8

References 6AI J"?8 > A ( # I ( J ( )! 5 P ( ?"? 6I"?8 1 I(

%" # !(

# ! 2 ( K 1 D 5 ( ?"?

6I 5 !8 3 I C ' 5 C( 5 - ( $ " +?!,( !M

6IN "8 N I- S(* # N )( $ ( E 1 5 N J >& + H ,½¾( ! +?",( M? 6IN5"8 N I- S( # N )( 5 5

S( C ( +?",( "M?? 6IN 5?8 N I- S( # N )( = 5( A( ??

#

2 # !( K )

6I5 ?8 N I- S 5 5

S( E * & + ,)( )! +??,( M! 6N?8 > N ( ( " +??,( ?M

" !

6N=B8 > N 1 = = ( 1

) Y( " &! (


572

Chapter 6


6N !B8 # N )( ( 2 *

* ( +?!B,( M 67 !8 # 7 ( ) ( ?"MB 6 B8 7 (

'! - +?!,(

2 !* P $( $$( I ( ?

B

6I55! 8 K # ( I ( K 5 ( > 5 ( 3 ( ( " +?! ,( BM! 6 ?8 I ( "

!( I =

6 B8 I ( "

( ??

" ( 5 P ( BB

6=?8 1 = ( $ ( " ! # +??,( ?M 6= 8 K = 0 ( E

( BM

+?

6= !8 K = 0 ( > ( M ! &&( N 7 - ( >( # +?!B,

,(

# *

63 ?8 K 3 ( " ! 2 " !( N ( ' # )( ?? 6# # 8 = # # #( 2 5 ( 1) D 5

( ? 6# ?8 A # ( 3 \ ( !M"

5 &1 !(

)! +??,(

6# !8 # C(

( +?!,( !!M? 6JBB8 # J ( # )( BBB

! 2 " !* # ( '

6B8 # S O 9 S( 5 ( ' G9( BB

! " !( 7

65 8 = 5( AC C G -

( ) +? ,( ?M 65!8 K 5-( !

! ! ( J N +>2 7 =,(

( ?!( !MB! 65 ?8 * 5 5

S( 5 + ,)( " +??,( ?M!


Section 6.5

573


65 !B8 K = 5

( 5 ( BMB ! !( >2 ( = = ( J 5( K 5L ( A +?!B, 61 J?8 E M" +??,( !!M?


*

(

" !

574

6.6

Chapter 6


MATROIDAL METHODS IN GRAPH THEORY ' $( ) # 2 A N% >& & 5

N # ' $ I : 7 E ! I # " ' # ? >& # I C B 1 ( 1 ( 5

) I # ' # I

Introduction > (

: ( E ( C

- ) 1 +?!MBB, 6!?82 .$ ) ) & )) & % )

/ )

6.6.1 Matroids: Basic Definitions and Examples

&

= 1 +?B!M?"?,

% ? 618 DEFINITIONS

2 9 % +9 , + 9 , +9 , ) ) ( ½ ¾ ) +9 , ½ ¾( ) ¿ +9 , ¿ +½ ¾ ,

7: ( +9 , +9 , ))


Section 6.6

575

Matroidal Methods in Graph Theory

2 ) ) (

2 2

+ , &

9½

G +9½ , +9¾ ,

G+ , 9¾

( 9½ 4 9¾(

9½

9¾

) -+9 ,(

9

) ,+9 ,

REMARK

2

$ % ) EXAMPLES

2

0 & )

$* 3* " #E$N 9

+ ,( %* "

E'JN 5> +9 ,

9

I$I'$5 +9 ,

$JN>>JN>J 5>5( ,+9 ,

A5>5 -+9 ,

C + , 2 C

7 2

) 0

C 2 C )

&

2 4 , H

C 2 C ,

4 2 4 4 ,

+ ,(

2 " 6 8(

9 0

& 0 % 7

) "( @

+B , ,

2

3 9 ) +9 , 4 +9 , 4 9 4 9 + ½ , 4 9 + ¾,( ½ ¾ 7 -+9 , 4 ( 9 4 9 608( 0 &

B B B


B B

B B

B B

B B

576

Chapter 6


1 1 2

3

4

2 G

3

1

5

4 G 2

6 5

6

Figure 6.6.1 9 ½ " %*" " 9

2 $ ) ( % 9 % % $* / * " I

I

9 4 9 + ,

)

9 4 9 608 & 0 %

)

) 7 +,( %

) 7 +,

) %

FACTS

$ ( 9

2 : @ 2 $ 9 ( 9 4 9 + , 2 +1 * $ 618,

) ) ) : 2 + , & < +

, + ,< +

, ) ) 9 ) ( ) ( (

:

2 :

) 2 $ 9 ( 9 2 9 9 ) ) ) ( & )

B


Section 6.6

577


6.6.2 Alternative Axiom Systems # ) C ) 0 & & E ) ( & ( 6E&?8 (

% $ )

"" 3 ) , $

:; ,< :; ) ) , , +, ,< :; , (

,

& 6 8 6


8

580

Chapter 6


2 2 (

) %& % ( (

$* /* " * )

)

9 9

"

9

9

9

9

9

9

REMARKS

2

) )

)

2

# $ ( & ) 9 + , 9 + , ( &( &

6.6.5 Matroid Union and Its Consequences ( ) J1 6J 8( :

DEFINITION

2 9½ 9 9

) C C C C ,+9 , +5 7 ", $ ) 9 . 9 . . 9 FACTS

2 3 9 9 9 )

) C C C C ,+9 , ( %

2

$

9

- ( -

9

. 9 . . 9

+ , H 2

Covering and Packing Results

- ) - ( 7 ) ) 7 B ( ( J 7 B


Section 6.6

581


FACTS

2

9 ' 9 ) ( ) '+ , H +9 , '+9 , 2 6> 8 9 ' +9 ,

( ) +9 ,( '+ , +9 ,(

6> 8

2

'

6 ( J 8 9

( + ,( ) 9 0 + , )

#

' #

2

#

6 8 ( ) + ,(

#

) ' 9

'

+ 6 8, + ,

2

6> 8 3 ) 4

) )

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4 +)

) , : 2 + ,

% 4

)

9 +

, 7 # + ,( ) 9 +# , +

,

6.6.6 Fundamental Operations N ) ) ( ( % ) ) ( ( C > C DEFINITIONS

9"

9)"

9 + " , 9+ " , 9 " 2 B 9 B ) ) 9 ) :

B B 4 9 2 ! ! ) ) ) :

2

( (

7 :

9


9)

9 ( 9 9)

582

Chapter 6

$* #

+9 , "

"

"

C +9 , " C , +9 ,

+9 , "

"

9½ 9¾ (

+9½ , +9¾ ,

2

+9½ , +9¾ ,

4

-+9 " ,

C½ C¾

" 9½

2

C 4 , +9 ,

)

9

C +9 , "

+9 ,

2

9

2

+9 ,

9)" (

,

+9 , "

"*

$ "

5

9 " (


2

C , +9 ,

9¾ (

+9½ , +9¾ ,

4

EXAMPLES

2 9 + , 4 9 + ,( ) ) 2 9 + ,) 4 9 + ),( ) ) )

(

( )

2 @ 4 @

½

2 @ ) 4 @ ½

½

,4

,4B

@

@ ½

½

@ ) 4 @

½

4

2 9 608 & ) )

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0

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& ) ) )

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$

½

¾

9 608)

& 9 (

9 + ½, 9 + ¾,

½ ¾ # ( ½ & ½ ¾ & ¾ ( 9 + ½ , 9 + ¾ , ) ) ½ ¾ ( ) ½ ¾

) ) - 9

FACTS $

2

(

9)" ,

+

9 ( 9½ (

4

9 "
&

@

@

)

@

@ 7 7

I

>&

@ @ 7 7

@ 7 7 9

+ , 9 + ,

+

,

+

@ 7 7 9 9

,

2

)

CONJECTURES

9 )


588

Chapter 6


1! + * I 9 6 !8, 7 % % ( % &

) 1! 7 % % ( /

) ( % &

/ REMARKS

2

7 " )) (

Æ 7 ? 7 " & 9 ) 7 " ) &)

2

7 ) ( > ( &

7 +>, ) % ( ( G 6GBB8 7 +,( ) & &

* I 9 > & 5 I 9 I ) ) 1 61B8 ) ( ( 1 61B8

6.6.10 Wheels, Whirls, and the Splitter Theorem 6 8 % ) ) ) - 5): ( C 6 8 5 ( C ( ) 5 65"B8 ( ( ) J

6J"8 DEFINITIONS

2 7 ( 0 ) & 9 ) + , &

2

7 ( 0 0 0 & ( -

2 3 9

B

B

)

9

B

9

2

/ - ½

/ ( ' ( ' - ' H

EXAMPLES

2

7 0 ( ( 9 +0 , 0 9 +0 , 0


Section 6.6

589

Matroidal Methods in Graph Theory 1 5

1

6

1

(a)

(b)

6

5

(c)

6

5

2

3

2

4

Figure 6.6.5 :;

3

2

3

4

0

4

" :$; 9 +0 , " :; 0

2 A 7 7 & - 2 : - ) - & 6C C 8 +, C & ) ) C

& C (

7

FACTS

2 6 8 3 : 2

)

+ , 7 ( +

,

) )

-

2 6 8 3 9 ) : 2

+ , 7 +

,

9

9 9)

9 +0 , 0

9

(

2 + 5 65"B8, 3 9 B )

B

9 ( +B , ( 9 5 B 4 9 +0 ,( 4 0 ( 9 0

< B 9 9 +0 ,

: 9 9 9

9 4 9 < 9 4 B < ( ( 9

9

) 5 ( I 6I "8 + 6I E&?8,( 5 * 5 ) $ - 7 ) ( % 5

2 61 B8 3

7

!

)

) 4 !( ! #L

Figure 6.6.6


# .' $ *""7

!

590

Chapter 6

2 6=8 3 ) 4


2 65"B8 3 9 ) ) 9 9 4 7

2

3 4 0

2

)

6E&"?8 3

9

7

0

)

0

+ ,

7 !< +

, ' ( ) ) 9 & )

Figure 6.6.7

- #-*

) 6?8 ( ( (

2

6E E&?8 7

B

2

( B

0

6N E E&P?!8 7 ( B B

@

@

+

9

, 9 + , 9 +0 , 0

-

6.6.11 Removable Circuits # ' ) '

C ' 4 = ( #* DEFINITIONS

2 ' & )

) '

'

2 7 ' (


' 9 &

9

Section 6.6


591

FACT

2 +#, 6#!8 $ '

' H ( ) #* + , ½ +' H , + , &

) )

& )

2 63E&??8 3 9 ) ) 9 $ +9 , +9 , H +9 ,( 9 9 9 +9 , 4 +9 , $ ( + , 4 +9 , +9 , +9 , H ( 9 ) 2 63E&??8 3 9 ) ) 9 $

+9 ,

"

+9 , H +9 , 4 +9 , H ( +9 , H +9 ,

9 9 9 +9 , 4 +9 , & ) )

2 3 ) ) $ + ,

+ , + ,( )

$ (

+ , + ,( ) 2 3 ) ) 5

+ ,

"

+ ,

( + , + ,

2 6 #?!8 3 )

$

( 9 )

2 6#BY8 3 ) $

( ) ) 7 ( #* +7 B, ) ) )

2 65 ?"8 3 )

%

) >& >? ) C & C ) & )

2 6 K??8 3 9 ) ) % $ 9

) 7 7 ( 9 ) +9 , 4 +9 ,


592

Chapter 6


REMARKS

2 7 ( )

)

2 7 ) ) ( (

( ) ) ( ( (

2 = )) ) ) - >

) EXAMPLES

2 63E&??8 I 2 )

& ! " ? B< ) ! " ? B( & E& 9 ) ) C 9 + , C A 9 + , ) ) ) C 7 B

) ' 4

2 K- 6K"B8 ( ( ) + 6K"B8, = ))* : +?, ) % 7 "+, 2 6 #?!8 7 ) ( 7 "+),

) ) ) 9( # 7

2 7 ( @ ( H ( )

(a) G 1

(b) G 2

Figure 6.6.8 8

+

9

,

9

+ ,

2$*

PROBLEMS

4:;! 6 K??8 $ $ ) $ ) Y 4:;! 6 K??8 $ 9 ) (


9

) Y

Section 6.6

593


6.6.12 Minimally '-Connected Graphs and Matroids 7 ' ( ' ' ' 7 ' ( DEFINITIONS

2 7 ' ( ' '

'

2 7 ' ( ' 9 '

2 3

9

9

)

9

9

'

EXAMPLES

2

$ , ' (

' 7 ( - 0

0

2

7 7 )

FACTS FOR ARBITRARY CONNECTIVITY

2 6#!8 7 & '

'

(

'

2

6#!?8 7 ' ( )

'

'

'

'

+' , + , H ' '

2

6E&")8 7 ' ( ) + , + , H ' REMARKS

!

7 !B ' 4 ) N 6N !8

6 "8( ( ' 4 ) = 6= ?8 7 ! )

!

) 7 !( ) 7 ? )

( : ) 7 !B

!

6#? 8 ) - ) 9 &

)

& 7 ? ) ) '


594

Chapter 6


FACTS FOR SMALL CONNECTIVITY

2 $ 9 9 4 9 + ,

( ) +9 , 9 & 2 6N !( "( = ?8 7 ' (

'

'

2 6E&"( E&")8 7 ' ( + , +

,

9

2 61?"8 3

9

9

2 6= ?8 3

+9 , H + '

)

)

+ , +

,

)

'

' < , ' '

9

9

A &)( G - * 1* (

+?!!,( M

6A !?8 > A &)( E * C (

+?!?,( !MB

6AE&?8 A - K E& (

( M + J 1 ,( I ) ' ( ??

6I "8 I I ( .# # 9 A # /( N ( J ' ( ?" 6I E&?8 I I K E& ( >& * 1 1 ( +??,( BMB 6I"8 1 = I ( E ( +?!?,( ?M??

B

6N E E&P?!8 N ( A E - ( K E& ( N P ( ' )

( ! +??!,( M?? 6> 8 K > ( 3 * J 1 ( %! ( ! ?A +? ,( !M! 6>?8 >U ( E & ( " B +??,( !M

61B8 K 1 ( A * 9( " +BB,( MB


596

Chapter 6


6GBB8 K 7 ( # = ( G ( &

7 +,M ) (

!? +BBB,( !M?? 61B8 K ( # = ( 1 ( A : ( " +BB,( "!

6 K??8 3 A K- ( ) ) ( +???,( ?M

6 #?!8 3 ( K = ( 5 # ( )

( ! +??!,( BM 6?8 A 5 )(

( )! ?M! 6=8 N 1 = (

- ( M!

B +??,(

? +?,(

6= ?8 = ( ' L)

( " +? ?,( !M""

C

L

6= ?8 = ( O C

L ( $ " +? ?,( M 6$1"8 $- 3 1 )( 1 ( )! ! +?",( M

& "

6K"B8 A K- ( )

( 9 +?"B,( "M? 6GB8 G G - ( 5 ) d ) ( 7 +?B,( !M" 63"8 3C ( ) ( +?",( M 63 8 3 ( 5 ( +? ,( "!M! 63!?8 3 ( E : ( BM!

& !

" " !

+?!?,(

63E&B8 # 3 K E& ( ) C ( ! +BB,( B?MB 6#!8 1 #( >-

( +?!,( ?M

C

L

6#!8 1 #( GC 1 ( " +?!,( "!MB

$

6#!?8 1 #( I % ( M? ! ! + A A )S,( I ) ' ( ?!?


Section 6.6

597


6#? 8 1 #( E

( !* :

! ! " 3 9 + N # - S( P 5S ( 5CU ,( KS A # 5 ( A ( ?? ( M? 6#BY8 5 # ( I &

( )

6#BY8 5 # ( I ) ) ( )

6J 8 I 5K J1 ( > 9 % ( +? ,( MB

6J 8 I 5K J1 ( (

2 # ! +$ 5 ( ,( N ( ( ? ( M 6J"8 5 J

( C ( +?",( ?M! 6J '??8 P J 3( > I ( K ' ( ) ( )! ?!R?" +???,( M 6E E&?8 A E - ( K E& ( ( ) ( ! +??,( ?M! 6E&"8 K E& ( E C ( ?B +?",( B!M 6E&"8 K E& ( E ( ?MB"

6 ,52 9 +?",(

6E&")8 K E& ( E ( +?",( !M"

!

6E&"?8 K E& (

( +?"?,( ?MB

( E& ' ( ??

6E&?8 K E& (

6E&B8 K E& ( E ) ( ! !* 9;;< + K 1 = ,( 3 # 5 3 J ""( I ) ' ( BB( ??M? 6 !8 I ( I ) ( ( +J ( 5 ?!B,( P ( ( ?!( ?M

& "

6 5BY8 J ) N 5 (

[[ 1* 9( ( 65" 8 5 9(

2 &" " "( 1 ( ?"

65!?8 N 5 ( # +?!?,( ?M!


7 +,(

598

Chapter 6


65"B8 N 5 ( N ( +?"B,( BM?

"

65 ?"8 5 ( .5 5- > I /( N ( ' 3 ( ??" 6"8 1 ( $( ! +?",( M B 6"8 1 ( $$( "" +?",( M! 6?8 1 ( # ( !M

""

!

!

+??,(

6 8 1 ( E ) ( +? ,( MB 6 8 1 ( 3 ( +? ,( M!

%! ( !

6 8 1 ( I (

" +?

L 61!8 G 1( ') > 5C G - ( +?!,( "BM" 61 B8 G 1( A - C = P ( M 61??8 N 1 (

( BM" 618 = 1 ( (

?A

,( BM

)

+? B,(

% ! " !

+???,(

+?,( M

618 = 1 ( E ) ( ! +?,( B?M

61?"8 = 1( E & ) (

! +??",( "M?!

61?!8 3 1( ) ) (

+??!,( B!M 61BB8 3 1( >& ) ( )! +BBB,( ??MB"


599

Chapter 6 Glossary

GLOSSARY FOR CHAPTER 6 "?% 3 M

2 B & 0

( )

9 (

F # "2

%

9 &(

0"

0

4

C

&(

F

* 2 M % 2

% )

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0

B

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* 2 2

0

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6

6

2

%$+ ,

#2

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# 2 2

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2

M

(

(

M (

%$+ ,

2

$ "" M % 2

) ) &

$ @2 $

& 9 %

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92

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M 2 ) )

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)9 2 )

1

(

# +4 ,

2*2

4

(

)9 # 1 (

(

4

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4

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M

2

)

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M &

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M 2

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& )

1* $ 2 : ) % ) 4 4 H H H 1%*% " M

% 2 % ( %

( +% ,


&

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I

600

Chapter 6

1%*% ½2


I

1%*% 2

I

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)

1%*%

2 ) ) (

*% * M

2

+C

0, 9

&

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Minimum and Maximum Imbeddings

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Voltage Graphs

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Voltage Graphs

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672

Chapter 7


DEFINITIONS

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Handbook of Graph Theory

Handbook of Graph Theory

Handbook of graph theory

Graph theory

Graph theory

Graph Theory

Graph Theory

Graph Theory

Graph Theory

Graph Theory

Graph Theory

Graph theory

Graph theory

Graph Theory

Graph Theory

Graph theory: proceedings of the Conference on Graph Theory

A Textbook of Graph Theory

Graph Theory and Applications

Graph Theory Singapore 1983

Introduction to graph theory

Algorithmic graph theory

Graph Theory with Applications

Introduction to graph theory

Algebraic Graph Theory

Graph theory and applications

Graph Theory 3

Graph Theory and Applications

Graph Theory: Conference Proceedings

Convexity and graph theory

Graph Theory and Applications

Algorithmic graph theory

Handbook of Graph Theory