Rainbow domination in the lexicographic product of graphs

A k-rainbow dominating function of a graph G is a map f from V (G) to the set of all subsets of {1, 2, . . . , k} such that {1, . . . , k} = ⋃ u∈N(v) f(u) whenever v is a vertex with f(v) = ∅. The k-rainbow domination number of G is the invariant γrk(G), which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by a k-rainbow dominating function. We focus on the 2-rainbow domination number of the lexicographic product of graphs and prove sharp lower and upper bounds for this number. In fact, we prove the exact value of γr2(G ◦ H) in terms of domination invariants of G except for the case when γr2(H) = 3 and there exists a minimum 2rainbow dominating function of H such that there is a vertex in H with the label {1, 2}.