A note on isolate domination

A set S of vertices of a graph G such that (S) has an isolated vertex is called an isolate set of G. The minimum and maximum cardinality of a maximal isolate set are called the isolate numberi0(G) and the upper isolate number I0(G) respectively. An isolate set that is also a dominating set(an irredundant set) is an isolate dominating set (an isolate irredundant set). The isolate dominationnumber 0(G) and the upper isolate domination number Γ0(G) are respectively the minimumand maximum cardinality of a minimal isolate dominating set while the isolate irredundance numberir0(G) and the upper isolate irredundance number IR0(G) are the minimum and maximumcardinality of a maximal isolate irredundant set of G. The notion of isolate domination was introduced in [5] and the remaining were introduced in [4]. This paper further extends a study of these parameters.