A subset $D$ of the vertex set $V(G)$ of a graph $G$ is called a \emph{dominating set} of $G$ if every vertex in $V-D$ is adjacent to a vertex in $D$. The minimum cardinality of a dominating set is called \emph{the domination number} and is denoted by $\gamma(G)$. A dominating set $D$ such that $\delta(<D>)=0$ is called an \emph{isolate dominating set}. The minimum cardinality of an isolate dominating set is called \emph{the isolate domination number} and is denoted by $\gamma_0(G)$. In this paper we investigate the properties of the graphs for which $\gamma_0=n-\Delta$.