Doubly Isolate Domination in the Corona of Graphs

The doubly dominating set of a graph $G$ is a set of dominating set $S\ss V(G)$ such that both $\ll S\rr$ and $\ll V(G)\ssm S\rr$ have an isolated vertex. The cardinality of a doubly isolate dominating set with minimum cardinality is called doubly isolate domination number and is denoted by $\00(G)$. In this paper, we characterize the doubly isolate dominating set for the corona and lexicographic product of two graphs.