Doubly Isolate Domination in Lexicographic Product of Graphs

For given positive integers a, b, and c such that 2\leq a\leq b\leq c, we show that there exists a connected graph G such that \gamma(G)=a, \gamma_0(G)=b, and \gamma_{00}(G)=c where \gamma is a domination number and \gamma_0 is an isolate domination number. Moreover, we characterize the doubly isolate dominating set for the lexicographic product of two graphs.