Vertex criticality with respect to isolate domination

Abstract

A set S of vertices of a graph G is called a dominating set of G if every vertex in V(G) - S is adjacent to a vertex in S. A dominating set S such that the subgraph 〈S〉 induced by S has at least one isolated vertex is called an isolate dominating set. The minimum cardinality of an isolate dominating set is called the isolate domination number and is denoted by γ0(G). This concept was introduced in [I. Sahul Hamid and S. Balamurugan, Isolate domination in graphs (Communicated)] and further studied in [I. Sahul Hamid and S. Balamurugan, Extended chain of domination parameters in graphs, ISRN Combin. 2013 (2013), Article ID: 792743, 4 pp.; Isolate domination and maximum degree, Bull. Int. Math. Virtual Inst. 3 (2013) 127–133; Isolate domination in unicyclic graphs, Int. J. Math. Soft Comput. 3(3) (2013) 79–83]. This paper studies the effect of the removal of a vertex upon the isolate domination number.