Isolate domination in Unicyclic graphs

A subset $D$ of the vertex set $V(G)$ of a graph $G$ is called a dominating set of $G$ if every vertex in $V-D$ is adjacent to a vertex in $D$. A dominating set $D$ such that $$ has an isolated vertex is called an isolate dominating set and the minimum cardinality of an isolate dominating set is called the isolate domination number of $G$ and is denoted by $gamma_0(G)$. In this paper we characterize the unicyclic graphs in which the order equals the sum of the isolate domination number and its maximum degree.