computational fluid dynamics

Pattern Recognition Letters 19 Ž1998. 787–791

Domination in fuzzy graphs – I A. Somasundaram a , S. Somasundaram

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Department of Statistics, Manonmaniam Sundaranar UniÕersity, TirunelÕeli 627012, India Department of Mathematics, Manonmaniam Sundaranar UniÕersity, TirunelÕeli 627012, India Received 13 August 1997; revised 26 February 1998

Abstract In this paper we introduce the concepts of domination and total domination in fuzzy graphs. We determine the domination number g and the total domination number g t for several classes of fuzzy graphs and obtain bounds for the same. We also obtain Nordhaus–Gaddum type results for these parameters. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Fuzzy graph; Domination; Total domination; Bipartite graphs; Independent subset

1. Introduction Rosenfeld Ž1975. introduced the notion of fuzzy graph and several fuzzy analogs of graph theoretic concepts such as paths, cycles and connectedness. Bhattacharya Ž1987. and Bhutani Ž1989. investigated the concept of fuzzy automorphism groups. McAlester Ž1988. presented a generalization of intersection graphs to fuzzy intersection graphs. Mordeson Ž1993. introduced the concept of fuzzy line graphs and developed its basic properties. In this paper we introduce the concept of domination in fuzzy graphs. For graph theoretic terminology we refer to ŽHarary, 1969..

2. Definitions We review briefly some definitions in fuzzy graphs and introduce some new notations. Let V be a finite nonempty set. Let E be the collection of all two-element subsets of V. A fuzzy graph G s Ž s , m .

is a set with two functions s : V ™ w0,1x and m : E ™ w0,1x such that m Ž x, y4. ( s Ž x . n s Ž y . for all x, y g V. Hereafter we write m Ž xy . for m Ž x, y4.. Let G s Ž s , m . be a fuzzy graph on V and V1 : V. Define s 1 on V1 by s 1Ž x . s s Ž x . for all x g V1 and m 1 on the collection E1 of two element subsets of V1 by m 1Ž xy . s m Ž xy . for all x, y g V1. Then Ž s 1 , m 1 . is called the fuzzy subgraph of G induced by V1 and is denoted by ² V1 :. The order p and size q of a fuzzy graph G s Ž s , m . are defined to be p s Ý x g V s Ž x . and q s Ý x y g E m Ž xy .. Let s : V ™ w0,1x be a fuzzy subset of V. Then the complete fuzzy graph on s is defined to be Ž s , m . where m Ž xy . s s Ž x . n s Ž y . for all xy g E and is denoted by K s . Let G s Ž s , m . be a fuzzy graph on V and S : V. Then the fuzzy cardinality of S is defined to be Ý Õ g S s Ž Õ .. The complement of a fuzzy graph G denoted by G is defined to be G s Ž s , m . where m Ž xy . s s Ž x . n s Ž y . y m Ž xy ..

0167-8655r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 8 6 5 5 Ž 9 8 . 0 0 0 6 4 - 6

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A. Somasundaram, S. Somasundaramr Pattern Recognition Letters 19 (1998) 787–791

An edge e s xy of a fuzzy graph is called an effectiÕe edge if m Ž xy . s s Ž x . n s Ž y .. N Ž x . s  y g V < m Ž xy . s s Ž x . n s Ž y .4 is called the neighborhood of x and N w x x s N Ž x . j  x 4 is the closed neighborhood of x. The degree of a vertex can be generalized in different ways for a fuzzy graph. The effectiÕe degree of a vertex u is defined to be the sum of the weights of the effective edges incident at u and is denoted by dEŽ u.. Ý Õ g NŽ u. s Ž Õ . is called the neighborhood degree of u and is denoted by dN Ž u.. The minimum effectiÕe degree d E Ž G . s min dEŽ u.< u g V Ž G .4 and the maximum effectiÕe degree DE Ž G . s max dEŽ u.< u g V Ž G .4 . In a similar way the minimum neighborhood degree and the maximum neighborhood degree denoted by d N and DN , respectively, can also be defined. We observe that these concepts reduce to the usual degree of a vertex in crisp case. A fuzzy graph G s Ž s , m . is said to be bipartite if the vertex set V can be partitioned into two nonempty sets V1 and V2 such that m Ž Õ 1Õ 2 . s 0 if Õ 1 ,Õ 2 g V1 or Õ 1 ,Õ 2 g V2 . Further if m Ž uÕ . s s Ž u. n s Ž Õ . for all u g V1 and Õ g V2 then G is called a complete bipartite graph and is denoted by K s1 , s 2 where s 1 and s 2 are, respectively, the restrictions of s to V1 and V2 .

3. Domination in fuzzy graphs The earliest ideas of dominating sets date back to the origin of the game of chess in India over 400 years ago in which placing the minimum number of a chess piece Žsuch as queen, knight, etc.. over the chess board so as to dominate all the squares of the chess board was investigated. The formal mathematical definition of domination was given by Ore Ž1962.. Cockayne and Hedetnieme Ž1977. published a survey paper on this topic in 1977 and since then over 400 papers have been published on this subject. The rapid growth of research in this area is due to the following three factors. 1. the diversity of applications of domination theory to both real world and mathematical coverings or location problems. 2. the wide variety of domination parameters that can be defined.

3. the NP-completeness of the basic domination problem, its close and natural relationship to other NP-complete problems and the subsequent interest in finding polynomial time solutions to domination problems in special classes of graphs. Let G s Ž V, E . be a graph. A subset S of V is called a dominating set in G if every vertex in V _ S is adjacent to some vertex in S. The domination number of G is the minimum cardinality taken over all dominating sets in G and is denoted by g Ž G . or simply g . Let G be a graph without isolated vertices. A dominating set S of G is called a total dominating set if the subgraph ² S : induced by S has no isolated vertices. The minimum cardinality taken over all total dominating sets of G is called the total domination number of G and is denoted by g t . A dominating set S of a graph G is called an independent dominating set of G if no two vertices in S are adjacent. We now proceed to extend these concepts fuzzy graphs. Definition 3.1. Let G s Ž s , m . be a fuzzy graph on V. Let x, y g V. We say that x dominates y in G if m Ž xy . s s Ž x . n s Ž y .. A subset S of V is called a dominating set in G if for every Õ f S, there exists u g S such that u dominates Õ. The minimum fuzzy cardinality of a dominating set in G is called the domination number of G and is denoted by g Ž G . or g. Remark 3.2. 1. Note that for any x, y g V, if x dominates y then y dominates x and hence domination is a symmetric relation on V. 2. For any x g V, N Ž x . is precisely the set of all y g which are dominated by x. 3. If m Ž x, y . - s Ž x . n s Ž y . for all x, y g V, then obviously the only dominating set in G is V. The above definition of domination in fuzzy graph is motivated by the following situation. Let G be a graph which represents the road network of a city. Let the vertices denote the junctions and the edges denote the roads connecting the junctions. From the statistical data that represents the

A. Somasundaram, S. Somasundaramr Pattern Recognition Letters 19 (1998) 787–791

number of vehicles passing through various junctions and the number of vehicles passing through various roads during a peak hour, the membership functions s and m on the vertex set and edge set of G can be constructed by using the standard techniques given in ŽBobrowicz et al., 1990; Reha Civanlar and Joel Trussel, 1986.. In this fuzzy graph a dominating set S can be interpreted as a set of junctions which are busy in the sense that every junction not in S is connected to a member in S by a road in which the traffic flow is full. Example 3.3. 1. Since  Õ4 is a dominating set of K s for all Õ g V we have g Ž K s . s min x g V s Ž x .. 2. g Ž G . s p if and only if m Ž xy . - s Ž x . n s Ž y . for all x, y g V. In particular g ŽK s. s p. 3. g Ž K s1 , s 2 . s min X g V1 s Ž x . q min y g V 2 s Ž y .. For the domination number g the following theorem gives a Nordhaus–Gaddum type result. Theorem 3.4. For any fuzzy graph G, y q g ( 2 p where g is the domination number of G and equality holds if and only if 0 - m Ž xy . - s Ž x . n s Ž y . for all x, y g V. Proof. The inequality is trivial. Further g s p if and only if m Ž xy . - s Ž x . n s Ž y . for all x, y g V and g s p if and only if s Ž x . n s Ž y . y m Ž xy . - s Ž x . n s Ž y . for all x, y g V which is equivalent to m Ž xy . ) 0. Hence g q g s 2 p if and only if 0 m Ž xy . - s Ž x . n s Ž y .. I Definition 3.5. A dominating set S of a fuzzy graph G is said to be a minimal dominating set if no proper subset of S is a dominating set of G.

Proof. Let D be a minimal dominating set and d g D. Then Dd s D _  d4 is not a dominating set and hence there exists x g V _ Dd such that x is not dominated by any element of Dd . If x s d we get Ž1. and if x / d we get Ž2.. The converse is obvious. I Definition 3.7. A vertex u of a fuzzy graph is said to be an isolated Õertex if m Ž uÕ . - s Ž u. n s Ž Õ . for all Õ g V _  u4 , that is, N Ž u. s B. Thus an isolated vertex does not dominate any other vertex in G. Theorem 3.8. Let G be a fuzzy graph without isolated Õertices. Let D be a minimal dominating set of G. Then V _ D is a dominating set of G. Proof. Let d be any vertex in D. Since G has no isolated vertices, there is a vertex c g N Ž d .. It follows from Theorem 3.6, that c g V _ D. Thus every element of D is dominated by some element of V _ D. I Corollary 3.9. For any graph without isolated Õertices g ( pr2. Proof. Any graph without isolated vertices has two disjoint dominating sets and hence the result follows. I Corollary 3.10. Let G be a fuzzy graph such that both G and G haÕe no isolated Õertices. Then g q g ( p, where g is the domination number of G. Further equality holds if and only if g s g s pr2. Example 3.11. For the fuzzy graph given in Fig. 1, g s g s 1, p s 2 so that g q g s 2.

The following theorem gives a characterization of minimal dominating sets which is analogous to the result of Ore Ž1962. in the crisp case. Theorem 3.6. A dominating set D of G is a minimal dominating set if and only if for each d g D one of the following two conditions holds. 1. N Ž d . l D s B. 2. There is a Õertex c g V _ D such that N Ž c . l D s  d4 .

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Fig. 1.

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A. Somasundaram, S. Somasundaramr Pattern Recognition Letters 19 (1998) 787–791

Definition 3.12. A set S of vertices of a fuzzy graph is said to be independent if m Ž xy . - s Ž x . n s Ž y . for all x, y g S. The following theorem gives a characterization of independent dominating sets. Theorem 3.13. If D is an independent dominating set of a fuzzy graph G then D is both a minimal dominating set and a maximal independent set. ConÕersely any maximal independent set D in G is an independent dominating set of G. Proof. If D is an independent dominating set of G, Dd s D _  d4 is not a dominating set for every d g D and D j  x 4 is not independent for every x f D so that D is a minimal dominating set and a maximal independent set. Conversely let D be a maximal independent set in G. Then for every x g V _ D, D j  x 4 is not independent and hence x is dominated by some element of D. Thus D is an independent dominating set of G. I The following theorem gives an upper bound for the domination number of a fuzzy graph. Theorem 3.14. g ( p y DN . Proof. Let Õ be a vertex such that dN Ž Õ . s DN . Then V _ N Ž Õ . is a dominating set of G so that g ( < V _ N Ž Õ .< s p y DN . I Remark. 1. The above inequality cannot be improved further. For example for the complete graph K s g s p y DN . 2. Clearly DE ( DN and hence g ( p DE . Definition 3.15. Let G be a fuzzy graph without isolated vertices. A subset D of V is said to be a total dominating set if every vertex in V is dominated by a vertex in D. Definition 3.16. The minimum fuzzy cardinality of a total dominating set is called the total domination number of G and is denoted by g t .

Theorem 3.17. For any fuzzy graph G, g t s p if and only if every vertex of G has a unique neighbor. Proof. If every vertex of G has a unique neighbor then V is the only total dominating set of G so that g t s p. Conversely suppose g t s p. If there exists a vertex Õ with two neighbors x and y then V _  x 4 is a total dominating set of G so that g t - p which is a contradiction. I Corollary 3.18. If g t s p then the number of Õertices in G is eÕen. Theorem 3.19. Let G be a fuzzy graph without isolated Õertices. Then g t q g t ( 2 p and equality holds if and only if 1. the number of Õertices in G is eÕen, say 2 n. 2. there is a set S1 of n mutually disjoint effectiÕe edges in G, 3. there is a set S2 of n mutually disjoint effectiÕe edges in G, and 4. for any edge xy f S1 j S2 , 0 - m Ž xy . - s Ž x . n s Ž y .. Proof. Since g t ( p and g t ( p, the inequality follows. Further, g t q g t s 2 p if and only if g t s g t s p and hence by Corollary 3.18, the number of vertices in G is even, say 2 n. Since g t s p, there is a set S1 of n disjoint effective edges in G. Similarly there is a set S2 of n disjoint effective edges in G. Further if xy f S1 j S2 then 0 - m Ž xy . - s Ž x . n s Ž y .. The converse is obvious. I

4. Conclusion The concept of domination in graph is very rich both in theoretical developments and applications. More than thirty domination parameters have been investigated by different authors, and in this paper we have introduced the concept of domination, total domination and independent domination for fuzzy graphs. Work on other domination parameters and domatic partition will be reported in forthcoming papers.

A. Somasundaram, S. Somasundaramr Pattern Recognition Letters 19 (1998) 787–791

References Bhattacharya, P., 1987. Some remarks on fuzzy graphs. Pattern Recognition Letters 6, 297–302. Bhutani, K.R., 1989. On automorphism of fuzzy graphs. Pattern Recognition Letters 9, 159–162. Bobrowicz, Choulet, C., Haurat, A., Sandoz, F., Tebaa, M., 1990. A method to build membership functions — Application to numericalrsymbolic interface building. In: Proc. 3rd Internat. Conference on Information Processing and Management of Uncertainty in Knowledge Based System, Paris. Cockayne, E.J., Hedetnieme, S.T., 1977. Towards a theory of domination in graphs. Networks 7, 247–261.

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