The isolate bondage number of a graph

A set \(D\) of vertices in a graph \(G\) is a dominating set, if each vertex of \(G\) that is not in \(D\) is adjacent to at least one vertex of \(D\). The minimum cardinality among all dominating sets in \(G\) is called the domination number of \(G\) and is denoted by \(\gamma(G)\). A dominating set \(S\) such that the induced subgraph by \(S\) has at least one isolated vertex is called an isolate dominating set. An isolate dominating set of minimum cardinality is called the isolate domination number and is denoted by \(\gamma_0(G)\). We define the isolate bondage number of a graph \(G\) to be the cardinality of a smallest set \(E\) of edges for which \(\gamma_0(G-E)>\gamma_0(G)\) and is denoted by \(b_0(G)\). In this paper, we initiate a study on the isolate bondage number.