  Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset
$S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one
vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by
$\gamma(G)$, is the domination number of $G$. A dominating set $S$ is an
isolate dominating set of $G$, if the induced subgraph $G[S]$ has at least one
isolated vertex. The isolate domination number, $\gamma_0(G)$, is the minimum
cardinality of an isolate dominating set of $G$. In this paper, we count the
number of isolate dominating sets of some specific graphs.
