Isolate domination in graphs

A set DD of vertices of a graph GG is called a dominating set   of GG if every vertex in V(G)−DV(G)−D is adjacent to a vertex in DD. A dominating set SS such that the subgraph 〈S〉〈S〉 induced by SS has at least one isolated vertex is called an isolate dominating set. An isolate dominating set none of whose proper subset is an isolate dominating set is a minimal isolate dominating set. The minimum and maximum cardinality of a minimal isolate dominating set are called the isolate domination number  γ0γ0 and the upper isolate domination number  Γ0Γ0 respectively. In this paper we initiate a study on these parameters.