ISSN: 2250-3005 - ijcer

International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2 Issue. 8Nonsplit Dom Strong Domination Number Of A Graph1, G. Mahadevan, 2, Selvam Avadayappan, 3, M. Hajmeeral1, Department of Mathematics Gandhigram Rural Institute- Deemed University Gandhigram – 624 3022, Department of Mathematics V.H.N.S.N. College, Virudhunagar-626 0013, Department of Mathematics B.S.Abdur Rahman University Vandalur, Chennai-600048AbstractA subset D of V is called a dom strong dominating set if for every v V – D, there exists u 1, u 2 D such that u 1 v,u 2 v E(G) and deg (u 1 ) ≥ deg (v). The minimum cardinality of a dom strong dominating set is called dom strongdomination number and is denoted by γ dsd . In this paper, we introduce the concept of nonsplit dom strong dominationnumber of a graph. A dom strong dominating set D of a graph G is a nonsplit dom strong dominating set (nsdsd set) if theinduced subgraph is connected. The minimum cardinality taken over all the nonsplit dom strong dominating sets iscalled the nonsplit dom strong domination number and is denoted by γ nsdsd (G). Also we find the upper bound for the sum ofthe nonsplit dom strong domination number and chromatic number and characterize the corresponding extremal graphs.1. IntroductionLet G = (V, E) be a simple undirected graph. The degree of any vertex u in G is the number of edges incident withu and is denoted by d(u). The minimum and maximum degree of G is denoted by δ(G) and ∆(G) respectively. A path on nvertices is denoted by P n. The graph with V(B n,n )={u 1 ,u 2 ,u 3 ,….u n ,v 1 ,v 2 ,v 3 ,…v n }and E(B n,n ) ={uu i ,vv i ,uv: 1 i n} is calledthe n-bistar and is denoted by B n,n . The graph with vertex set V(H n,n ) = {v 1 ,v 2 ,v 3 ,…v n ,u 1 ,u 2 ,u 3 ….u n } and the edge setE(H n,n ) = { v i , u j ,1 i n, n-i+1 j n} is denoted by H n,n . The corona of two graphs G 1 and G 2 is the graph G = G 1 o G 2formed from one copy of G 1 and |V(G 1 )| copies of G 2 where the i th vertex of G 1 is adjacent to every vertex in the i th copy ofG 2 .The Cartesian graph product G = G 1 □ G 2 is called the graph product of graphs G 1 and G 2 with disjoint vertex sets V 1and V 2 and edge set X 1 and X 2 is the graph with the vertex set V 1 x V 2 and u = (u 1 ,u 2 ) adjacent with v = (v 1 ,v 2 ) whenever[u 1 = v 1 and u 2 adjacent to v 2 ] or [u 2 = v 2 and u 1 adjacent v 1 ]. The book graph B m is defined as the graph cartesian productS m+1 x P 2 , where S m is a star graph and P 2 is the path graph on two nodes. The friendship graph or (Dutch windmill graph)F n is constructed by joining n copies of the cycle C 3 with a common vertex. The ladder graph can be obtained as theCartesian product of two path graphs, one of which has only one edge. A graph G is a called a (n x m) flower graph if it hasn vertices which form an n-cycle and n-sets of m-2 vertices which form m-cycles around the n-cycle so that each m-cycleuniquely intersects with n-cycle on a single edge.A (n, k)- banana tree is defined as a graph obtained by connecting one leaf of each of n copies of an k-star graph rootvertex that is distinct from all the stars. Recently many authors have introduced some new parameters by imposingconditions on the complement of a dominating set. For example, Mahadevan et.al [14] introduced the concept ofcomplementary perfect domination number.A subset S of V of a non-trivial graph G is said to be an complementary perfect dominating set if S is a dominating set and has a perfect matching. The concept of nonsplit domination number of a graph was defined by Kulli and Janakiram[5]. A dominating set D of a graph G is a nonsplit dominating set if the induced subgraph is connected. The nonsplit domination number γ ns (G) of G is minimum cardinality of a nonsplit dominating set. The concept of dom strongdomination number of the graph is defined in [16]. Double domination introduced by Haynes[18] serves as a model for thetype of fault tolerance where each computer has access to atleast two fileservers and each of the fileservers has directaccess to atleast one backup fileserver. Sampathkumar and Pushpalatha [15] have introduced the concept of strong weakdomination in graphs. A combination of the concepts of double domination and strong weak domination is the concept ofdomination strong domination where in for every vertex outside the dominating set, there are two vertices inside thedominating set, one of which dominates the outside vertex and the other strongly dominates the outside vertex. In thispaper we introduce the concept of non split dom strong domination number of a graph.Issn 2250-3005(online) December| 2012 Page 39