Bounds for Support Equitable Domination Number

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1 e-issn Volume 2 Issue 6, June 2016 pp Scientific Journal Impact Factor : Bounds for Support Equitable Domination Number R.Guruviswanathan 1, Dr. V.Swaminathan 2, M.Ayyampillai 3 1 Department of Mathematics, Maamallan Institute of Technology, Sriperumbudur. 2 Ramanujan Research Centre in Mathematics,Saraswathi Narayanan College, Madurai. 3 Department of Mathematics, Arunai Engineering College, Thiruvannamalai. Abstract Let be a simple graph. Let Support of denoted by and is defined as the sum of the degrees of neighbours of. Two vertices are said to be support equitable, if A subset of is called a support equitable dominating set of,if for any there exists a such that and are adjacent and and are support equitable. The minimum cardinality of a support equitable dominating set of is called support equitable domination number of and is denoted by In this paper, we obtain some bounds for Keywords Support, Equitable, Domination, support equitable dominating. I. INTRODUCTION Let be a finite simple undirected graph. The degree of a vertex in a graph is the number of edges of incident with and is denoted by. The maximum and minimum degrees of the vertices of are respectively denoted and (G). For any vertex v V, the open neighborhood and closed neighborhood of are denoted by and N[v] = N(v) {v} respectively. The support of a vertex in a graph is defined as the sum of the degree of its neighbours. A subset of is a dominating set of if every vertex in is adjacent with at least one vertex in.the minimum cardinality of a minimal dominating set is called the domination number of and is denoted by. For any graph theoretic terminology and notations we refer to Harry et.al[2] and for more details about parameters of domination number, we refer Haynes et.al[2]. Swaminathan et.al [6] introduced the concept of equitable domination in graphs,by considering the following real world problems; In a network nodes with nearly equal capacity may interact with each other in a better way. In this society, persons with nearly equal status, tend to be friendly. In an industry, employees with nearly equal powers form association and move closely. Equitability among citizens in terms of wealth, health, status etc is the goal of a democratic nation. In order to study this practical concept a graph model is to be created as follows: A subset of is called an equitable dominating set if for every v V-D there exist a vertex such that and, where and denotes the degree of a vertex and respectively. The minimum cardinality of such a dominating set is denoted by and is called the equitable domination number. In this paper, we use this idea to develop the concept of support equitable dominating set and support equitable domination number of a graph. Let ) be a simple graph. subset of a vertex set of a graph is called support equitable dominating set of if for any, there exists a, such that,.the minimum (maximum) cardinality of a support equitable All rights Reserved 11

2 set of is called support equitable domination number of (Upper support equitable domination of ) and is denoted by ) ( ).In this paper, we initiate a study on bounds for support equitable dominating number II. BOUNDS FOR SUPPORT EQUITABLE DOMINATION NUMBER Definition 2.1. Let be a simple graph. A subset of is called a support equitable dominating set of,if for any there exists a such that and are adjacent and and are support equitable. The minimum cardinality of a support equitable dominating set of is called support equitable domination number of and is denoted by. Definition 2.2. Let. The support equitable neighbourhood of, denoted by is defined as { } Definition 2.3. The maximum and minimum support equitable degree of a vertex in by are denoted Definition 2.4. Let be a given graph. Let be the graph constructed from as follows :,two vertices are in if and only if and are adjacent and support equitable in. is called support equitable associates of and is denoted by Theorem 2.5. Let be a graph without support equitable isolated vertices. (That is, given any there exists v such that and If is minimal support equitable dominating set of, then is a support equitable dominating set. Let. If is a support equitable isolate in, then has a support equitable neighbour in (Since has no support equitable isolates). If is not a support equitable isolate in then, as is minimal, there exists a support equitable private neighbour for u in. Therefore is a support equitable dominating set. Corollary 2.6. If has no support equitable isolate, then. Theorem 2.7. For any graph,. Let be a set of. Each vertex of has support equitably dominate, +1 vertices of. Therefore S can support equitably dominate, ( +1) vertices of G. Therefore n.therefore Therefore. Let V be a vertex of maximum support equitable degree. Then v support equitable dominates vertices. The vertices in V- dominates themselves. Therefore. Remark 2.8. When G =, = n, = 1 and Therefore the first inequality becomes an equality in the above theorem in the case of. Also in the case of ( ) ( ) ( ) Therefore ( ) ( ) ( All rights Reserved 12

3 The following examples gives classes of graphs G for which is Example 2.9. Let =. Then = 2 =. Also let G = Then = 2 =. Example Let be a regular graph of even order with. Attach pendants vertices, one each to vertices of. Let be the resulting graph. Then. Therefore.Then Therefore Example Let. Attach one pendant to each of the vertices forming one of the partition of. Let be the resulting graph. Then is support regular and. Then. Therefore. Example Let be the graph given below, and Support of each vertex in the central square is, support of each vertex in the surrounding square is. Therefore =. Theorem Let be a graph of even order and without support equitable isolates. Then if and only if is obtained from such that if and belong to two different cycles and then u and v are not support equitable. Further in any, adjacent vertices are support in OR is obtained from where each is either or All rights Reserved 13

4 that if and belong to and respectively and and are adjacent then and are not support equitable. Further, in any, support equitable adjacent vertices are support equitable in. Let,where n is an even. Case(i) Let for every. Claim for every Suppose the claim is not true. Then there exists vertex such that. By hypothesis Therefore Consider. The number of vertices of is, a contradiction, since. Therefore for every. Therefore for every Let. Since for every for every Therefore is a union of cycles say. Since. Therefore each is a. Then contains as a induced subgraphs and if and belongs to and respectively and if and are adjacent then and are not support equitable. Further in any adjacent vertices are support equitable in. Case(ii) Let for some. Let Then has pendant vertices. Since, each component of is the corona of a connected graph. that is. Suppose order of is. Then there exists a vertex i say of and the pendent, say attached to in is not support equitable with. Therefore each is either or. Therefore contains as induced subgraphs. If and belongs to and respectively ) and if and are adjacent then and are not equitable. For, if and have equitable supports then, a contradiction. Therefore and are not support equitable. Further, in any equitable in. The converse is obvious. support equitable adjacent vertices are support Remark Suppose.Then.Conversely, suppose. Let { }. Two vertices in are adjacent if and only if there are adjacent in and have equitable supports. Therefore has seven vertices and they form a cycle. Suppose has an additional edge. Then and. Therefore and cannot be adjacent in a contradiction. Therefore there is no additional edge in. Therefore Remark If is a connected graph, then need not be connected. For example All rights Reserved 14

5 se(g) = Then has an isolate vertex and hence not connected. If ) is connected then is connected. REFERENCES [1] F. Harary, Graph Theory, Addison - Wesley, [2] T. W. Haynes, Stephen T. Hedetniemi, Peter J. Slater, Fundamentals of domination in graphs, Marcel Dekker, [3] Teresa W.Haynes, Stephen T.Hedetniemi and Peter J.Slater, Domination in Graphs,Advanced topics (Marcell DekkerInc.1998) [4] E. Sampathkumar and H. B. Walikar, The connected domination number of a graph, Math. Phys. Sci., (13) (1979), [5] C.Y. Ponnappan, Studies in graph theory support strong domination in graphs, Ph.D., thesis Madurai Kamaraj university, (2008). [6] V. Swaminathan and K. M. Dharmalingam, Degree Equitable Domination on Graphs, Kragujevak Journal of Mathematics, 35(1), (2011). [7] Anitha, S. Arumugam, S. B. Rao and E. Sampathkumar, Degree Equitable Chromatic number of a a Graph, JCMCC, 75(2010), All rights Reserved 15