Nonsplit Dom Strong Domination Number Of A Graph

A subset D of V is called a dom strong dominating set if for every v V – D, there exists u1, u2 D such that u1v, u2v  E(G) and deg (u1 ) ≥ deg (v). The minimum cardinality of a dom strong dominating set is called dom strong domination number and is denoted by γdsd. In this paper, we introduce the concept of nonsplit dom strong domination number of a graph. A dom strong dominating set D of a graph G is a nonsplit dom strong dominating set (nsdsd set) if the induced subgraph is connected. The minimum cardinality taken over all the nonsplit dom strong dominating sets is called the nonsplit dom strong domination number and is denoted by γnsdsd(G). Also we find the upper bound for the sum of the nonsplit dom strong domination number and chromatic number and characterize the corresponding extremal graphs.