A Note on Domination Number of Quasi Complete Strong Fuzzy Graph

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 6 (2017), pp Research India Publications A Note on Domination Number of Quasi Complete Strong Fuzzy Graph 1 Narayana Boppana, 2 Sudhakaraiah A and 3 Mohiddin Shaw Shaik 1 Research Scholar, Department of Mathematics, Rayalasima Uniersity, Kurnool, A.P, India. 2 Department of Mathematics, S.V.Uniersity, Tirupathi, A.P, India. 3 Department of Mathematics, Narasaraopeta Engineering college, Narasaraopet, Guntur Dt., A.P, India. Abstract This paper is a study of Quasi Complete Strong Fuzzy Graph. In this study it is observe that, when G:(V, σ, µ) is a Quasi complete strong fuzzy graph with edge disjoint union of triangles, then all the triangles have a common vertex w and µ(wv ) i = min { σ(w), σ(vi) / vi is neighbor of w}. Domination and strong domination number of a Quasi Complete Strong Fuzzy Graph is one of the observation. In this paper, it is prove that If G is a Quasi complete strong fuzzy graph with n ( 3) vertices, then the dominating number of G is γ(g) = 1, and the strong dominating number is γ s (G) = min { μ(mn ) i where {m}is in dominating set D of G and ni, (1 i k) is strong neighbor of m, if and only if there exists a vertex m in G such that each edge of G lies on a triangle having m as a one of it vertex. Keywords: Fuzzy Graph, Quasi Complete Strong Fuzzy Graph, strong domination number. AMS Mathematic Subject Classification (2010): 05C72, 05C12, 05C38, 05C40, 05C INTRODUCTION: Rosenfeld introduced the notion of fuzzy graph and obtained several results by using graph theoretical concept. Bhutani and Rosenfeld have introduced the concept of strong edges and many researchers done a lot of work on fuzzy graphs with strong

2 1728 Narayana Boppana, Sudhakaraiah A and Mohiddin Shaw Shaik edges. In 19 th century a study on dominating sets in graphs started for solving problems in the game of chess. Domination in fuzzy graphs using effective edges was introduced by Somasundaram and Somasundaram [17]. Later Nagoor Gani and Chandrasekaran [16], O.T. Manjusha and M.S.Sunitha[5] were discussed and obtained several analogs results on this concept. They defined domination number as the minimum scalar cardinality of any dominating set [16] and using the weights[5] of vertices in dominating set D. The semi-complete nature of graphs were introduced and studied by Prof. I.H. Naga Raja Rao and S.V. Siva Rama Raju [3], which deals to tackle problems like sharing the information among the army authorities. Shaik Mohiddin shaw and Nagaraju Dasari [communicated] generalized the concept introduced by Prof. I.H. Naga Raja Rao and S.V. Siva Rama Raju. In this present study, we study about the domination and strong domination number of quasi complete strong fuzzy graph using the weights of strong edges in dominating set D. is derived which will be useful in fuzzy location problems in networks, Biology, defense etc., finally it is observe that it is prove that If G is a Quasi complete strong fuzzy graph with n ( 3) vertices, then the dominating number of G is γ(g) = 1, and the strong dominating number is γ s (G) = min { μ(mn ) i where {m}is in dominating set D of G and ni, (1 i k) is strong neighbor of m, if and only if there exists a vertex m in G such that each edge of G lies on a triangle having m as a one of it vertex. 2. PRELIMINARIES We review briefly some definitions in fuzzy graphs and notions which are presented in [10, 11, 12, 13, 14 ]. A fuzzy graph is a triplet G(V,σ,µ) where V the vertex set, σ is a fuzzy subset of V and µ is a fuzzy relation on σ such that µ(u, v) σ(u) σ(v) for all u, v V. We assume that V is finite and non empty, µ is reflexive and symmetric. In all the examples σ is chosen suitably. A weakest edge of a fuzzy graph G(V, σ, µ) is an edge with least membership value. The path length n of path P is a sequence of distinct vertices u 0,u 1,...u n such that µ(u i 1, u i ) > 0, i = 1,2,3...n. Here n is length of the path and µ value of a weakest edge in the path is its strength. Let G: (V, σ, µ) be a fuzzy graph. The length of the shortest path between two vertices vi and vj with membership grades σ(vi), σ(vj) is consider as the distance between two vertices. In this paper, it is consider that G(V, σ, µ) is as a connected fuzzy graph. A path P: u 0, u 1,...u n, length of the path P defined as sum of the membership grade of edges involved in that path. It is denoted by L(p). That is n L(p) = i=1 μ(u i 1, u i ). If n = 0 then L(p) = 0 and n 1, L(p) 0. Let v1 and v2 be any two vertices in a fuzzy graph G(V, σ, µ) and P = {pi / pi is a v1 v2 path, for i = 1,2,3 }. The sum distance [6] between v1 and v2 is defined as ds(v1, v2) = min {L(pi) / pi P, i = 1,2,3, }. If u 0 = u n and n 3, then P is called a cycle and a cycle P is a fuzzy cycle if it contains more than one weakest edge. The strength of connectedness between two vertices m and n is defined as the maximum of the strengths of all paths between m and n and is denoted CONNG(m, n). A fuzzy graph G(V, σ, µ) is connected if for every m, n in vertex set V CONNG(m, n) 0. A fuzzy

3 A Note on Domination Number of Quasi Complete Strong Fuzzy Graph 1729 graph G(V,, )is complete fuzzy graph if (m, n) = (m) (n) for all m, n V. The order and size of a fuzzy graph is defined as sum of membership values of vertices involved in that fuzzy graph and sum of the membership values of edges of the graph respectively. Let G(V, σ, µ) be a fuzzy graph with vertex set V and m, n are in V. Then we say that m dominates n in G if μ(mn ) = σ(m) σ(n). 3. QUASI COMPLETE STRONG FUZZY GRAPHS: In this section we studied about the Quasi Complete Strong Fuzzy Graph and obtain several theorems. Why we call this as a Quasi complete strong Fuzzy Graph, because all the edges in this graph are strong edges. 3.1 Definition: A fuzzy graph G(V,, ) is said to be a Quasi complete strong fuzzy graph if for any two vertices m, n there is a vertex s of G such that s is adjacent to both m and n in G and µ is such that µ(mn ) = σ(m) σ(n), for all edges in mn in G. 3.2 Note: 1. If µ(mn ) =1 for all the edges in G, then Quasi complete strong fuzzy graph is a semi complete graph. 2. A fuzzy graph with vertex set V= {m1, m2} and edge set E = { m 1 2 } with as fuzzy set on V and fuzzy set on E defined as (m ) 1 2 = ( m1) σ(m2) for all m1, m2 V is a complete fuzzy graph but not a quasi complete strong fuzzy graph. 3. A fuzzy graph with vertex set V= {v} with edge set E =, fuzzy set on V is trivially a quasi complete strong fuzzy graph and also complete. Throughout this paper we take non-trivial fuzzy graphs with at least three vertices and its membership grade greater than zero. 3.3 Example : Fuzzy graph G:(V, σ, µ) with vertex set V = {V1, V2, V3, V4, V5} and edge set {v 1 2, v, 2 3 v 3 4 v 4 5 v 2 4 v 2 5 v 1 5 } with membership grades σ(v1) =, σ(v2) = 0.9, σ(v3) = 0.9, σ(v4) =, σ(v5) = 0.9, and µ(v ) 1 2 =, µ(v ) 1 5 = µ(v ) 2 3 = 0.9, µ(v ) 3 4 =, µ(v ) 4 5 =, µ(v ) 2 4 =, µ(v ) 2 5 = 0.9 µ(v ) 1 5 = } is a quasi complete strong fuzzy graph.

4 1730 Narayana Boppana, Sudhakaraiah A and Mohiddin Shaw Shaik V 1 V V V 4 V Fig 3.3 :QCSFG(V,, ) 3.4 Observations: 1. Any Quasi complete strong fuzzy graph is non-trivial. 2. Every complete fuzzy graph with at least three vertices is Quasi complete strong fuzzy graph but inverse, that is a quasi complete strong fuzzy graph not a complete fuzzy graph. 3. Any two vertices of a Quasi complete strong fuzzy graph are at a distance at most two. But the converse of the stamen is not true. Let G: (V, σ, µ) be a Quasi complete strong fuzzy graph then for any two vertices u, v it have two paths either p1: u-v path or p2: u-w-v. 3.5 Definition: In a Quasi complete strong fuzzy graph G: (V, σ, µ) the distance between any two vertices u and v is defined as the min {L(p1), L(p2)}. It is denoted by dqcsfg(u, v). 3.6 Example: Fuzzy graph G:(V, σ, µ) with vertex set V = {V1, V2, V3, V4, V5} and edge set {v 1 2, v, 2 3 v 3 4 v 4 5 v 2 4 v 2 5 } with membership grades σ(v1) =, σ(v2) = 0.9, σ(v3) = 0.9, σ(v4) =, σ(v5) = 0.9, and µ(v ) 1 2 =, µ(v ) 2 3 = 0.9, µ(v ) 3 4 =, µ(v ) 4 5 =, µ(v ) 2 4 =, µ(v ) 2 5 = 0.9}, then dqcsfg(v1, v2) =, dqcsfg(v1, v3) = 1.7, dqcsfg(v1, v4) = 1.6, dqcsfg(v1, v5) =.8, dqcsfg(v2, v3) = 0.9, dqcsfg(v2, v4) =, dqcsfg(v2, v5) = 0.9, dqcsfg(v3, v4) =, dqcsfg(v3, v5) = 1.6, dqcsfg(v4, v5) =. 3.7 Theorem: A fuzzy graph G: (V, σ, µ) with at least three vertices is a Quasi complete strong fuzzy graph if and only if any two vertices u and v of G with membership grade σ(u) and σ(v) lie on the same triangle with µ(uv ) = minimum of σ(u) and σ(v) or they lie on different triangles having common vertex w with µ(uw ) = min {σ(u), σ(w)} and µ(wv ) = min {σ(w), σ(v)}.

5 A Note on Domination Number of Quasi Complete Strong Fuzzy Graph Theorem: A fuzzy graph G(V, σ, µ) is a Quasi complete strong fuzzy graph if and only if the distance between two vertices u and v is at most min {µ(uv ), µ(uw ) + µ(wv )}. 3.9 Theorem: A fuzzy graph G: (V, σ, µ) is a Quasi complete strong fuzzy graph if and only if it is the union of triangles in which the distance between any pair of vertices is at most min {µ(uv ), µ(uw ) + µ(wv )} Theorem: Let G: (V, σ, µ) be a Quasi complete strong fuzzy graph with edge disjoint union of triangles, then all the triangles have a common vertex w with σ(w) as membership grade and µ(wv ) i = min { σ(w), σ(vi) / vi is neighbor of w} 3.11 Theorem: Let G: (V, σ, µ) be a fuzzy graph with at least three vertices such that E(G) = E(T1) E(T2) E(T3) E(Tn) with E(Ti) E(Tj) =, where E(G), E(Ti) are edge sets of fuzzy graph G and i th triangle respectively. Then G is a Quasi complete strong fuzzy graph. Here all the triangles have a common vertex Theorem: A simple fuzzy graph G with at least three vertices and if there is an edge mn such that each vertex in V other than m and n is a neighbor to both m and n, then G is a quasi complete strong fuzzy graph. Proof: Let m1, m2 be two vertices in G with membership values σ(m) and σ(n). If {m1, m2} = {m, n}, there by the definition of quasi complete strong fuzzy graph there is vertex s with membership value σ(s) and it is adjacent to both m1 and m2 with µ(m ) 1 s = min {σ(m 1 ), σ(s), µ(sm ) 2 = min {σ(s), σ(m2)}. It gives that m1-s-m2 is a path with path length L(P2) = µ(m ) 1 s + µ(sm ). 2 Suppose one of m1, m2 is in {m, n}. Without loss of generality suppose m1 = m, then there exists a vertex s such that m1=m-s-n with path length L(P2) = µ(ms ) + µ(sn ). Let {m1, m2} {m, n}. Then there is a vertex s such that m1-s-m2 with path length L(P2) = µ(m ) 1 s + µ(sm ) STRONG DOMINATION IN QUASI COMPLETE STRONG GRAPH Ore and Berge introduced domination in graphs and later it is continued by Cockayne and Hedetniemi. For the notation we follow O.T. Manjusha and M.S. Sunitha [5]. According to Nagoor Gani and Chandrasekharan, a set D of vertices of graph G is a strong dominating set, if every vertex of V(G)-D is a strong neighbor of some vertex in D. They considered the set with minimum number of vertices as a strong dominating set and the scalar cardinality of minimum strong dominating set as strong dominating number of G. Nagoor Gani and etl., consider the membership value of the vertex when they spoke about dominating set and dominating number. Later it

6 1732 Narayana Boppana, Sudhakaraiah A and Mohiddin Shaw Shaik was optimized by O.T. Manjusha and M.S. Sunitha [5] by considering the membership value of strong edges incident on vertices in D of fuzzy graphs. Basing these arguments we continued the study on dominating nature of Quasi Complete Strong fuzzy Graph. Nagoor Gani and co-authors studied about semi-complete strong fuzzy graph. The difference between semi-complete and Quasi Complete Strong fuzzy graph is when n 4, the dominating set contains only one element so that the dominating number of this Quasi Complete strong fuzzy graph is Definition: Let G be a fuzzy graph with vertex set V. Let m, n in V. Then we say that m dominates n in G if μ(mn ) = σ(m) σ(n). A subset D of V is said to be dominating set in G if every m not in D, there exists n in D such that m dominates n. 4.2 Note: the vertices m and n are dominating each other when it is a strong edge in G. 4.3 Definition: The weight of a strong dominating set D is defined as W(D) = v D μ(uv ), where μ(uv ) is the minimum of the membership values of the strong edges incident on u. 4.4 Definition: The strong domination number of a fuzzy graph G is defined as the minimum membership value of strong dominating sets of G and it is denoted by γ s (G). A minimum strong dominating set in a fuzzy graph G is a strong dominating set of minimum weight. 4.5 Theorem: If G is a Quasi complete strong fuzzy graph with n ( 3) vertices, then the dominating number of G is γ(g) = 1, and the strong dominating number is γ s (G) = min { μ(mn ) i where {m}is in dominating set D of G and ni, (1 i k) is strong neighbor of m, if and only if there exists a vertex m in G such that each edge of G lies on a triangle having m as a one of it vertex. Proof: Let G is a Quasi Complete Strong Fuzzy Graphs with γ(g) = 1, and the strong dominating number is γ s (G) = min { μ(mn ) i where {m}is in dominating set D of G and ni, (1 i k) is strong neighbor of m. Since γ(g) = 1, there exist a vertex m in G such that m is adjacent to all other vertices in G. Let pq be an edge in G. If one of the vertices p or q is m, that is p = m (or q = m) then there exists a vertex j such that j is one of {p, m, q} which form a path in G. This shows that the edge mq lies on the triangle with vertices m, p, q. Suppose p and q are other than vertex m. Then there p and q are adjacent to m implies that the edge pq is an edge in the triangle whose vertices are m, p and q. Converse: Suppose v V(G)\{m}. Since G is a Quasi

7 A Note on Domination Number of Quasi Complete Strong Fuzzy Graph 1733 Complete Strong Fuzzy Graph, there is a vertex m1 in V(G)\{m} such that v is adjacent to m1. So that vm1 edge lies on a triangle containing m as a third vertex. This shows that v is adjacent to m. This is true for all v in G. This implies that m is adjacent to all v in G, so {m } is a dominating set. Therefore, γ(g) = 1, strong dominating of G is the minimum value of membership grades of the edges incident on m. Therefore, γ s (G) = min { μ(mn ) i where {m}is in dominating set D of G and ni, (1 i k) is strong neighbor of m. CONCLUSION Domination is well-to-do both in hypothetical and applications. There are almost more than thirty domination parameters have been investigated by many dissimilar authors, and in this study the concept of dominating and strong domination number has been studied. ACKNOWLEDGMENTS Authors would like to thank referees for their helpful comments for strengthen the paper. REFERENCES [1] Akaram.M, B. Davvaz, Strong intuitionistic fuzzy graphs, FOLOMAT26(1)(2013) [2] Bondy J. A. and Murty U. S. R. "Graph Theory with Applications", the Macmillan Press Ltd, (1976). [3] I.H. Naga Raja Rao and S. V. Siva Rama Raju Semi-complete Graphs,International Journal of Computational Cognition, Vol.7, No.3, 2009, PP [4] Kiran R. Bhutani, Abdella Battou, On M-strong fuzzy graphs, Information Sciences, 155(2003), PP [5] O.T. Manjusha, M.S. Sunitha, strong Domination in Fuzzy Graphs, Fuzzy information and Engineering, Elsevier (2015) 7: [6] Mini Tom and Sunitha M.S., Strong sum distance in fuzzy graphs, Springerplus, 4:214, DOI /s [7] Mini Tom and Sunitha M.S., Sum Distance in Fuzzy Graphs, Annals of Pure and Applied Mathematics, Vol.7. No.2, 2014, [8] Mathew J.K, Sunil Mathew, A characterization of fuzzy self-centered graphs, Annals of Fuzzy Mathematics and Informtics, vol

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