Some notes on the isolate domination in graphs

A subset S of vertices of a graph G is a dominating set of G if every vertex in V(G)−S has a neighbor in S. The domination numberγ(G) is the minimum cardinality of a dominating set of G. A dominating set S is an isolate dominating set if the induced subgraph G[S] has at least one isolated vertex. The isolate domination numberγ0(G) is the minimum cardinality of an isolate dominating set of G. In this paper we study the complexity of the isolate domination in graphs, and obtain several bounds and characterizations on the isolate domination number, thus answering some open problems.