On the weak Roman domination number of lexicographic product graphs

A vertex v of a graph G = (V, E) is said to be undefended with respect to a function f : V -> {0, 1, 2} if f(v) = 0 and f (u) = 0 for every vertex u adjacent to v. We call the function f a weak Roman dominating function if for every v such thatf (v) = 0 there exists a vertex u adjacent to v such that f (u) is an element of {1, 2} and the function f' : V -> {0, 1, 2} defined by f' (v) = 1, f '(u) = f (u) 1 and f '(z) = f (z) for every z is an element of V \ {u, v}, has no undefended vertices. The weight of f is w(f) =Sigma(v is an element of V(G))f(v). The weak Roman domination number of a graph G, denoted by gamma r(G), is the minimum weight among all weak Roman dominating functions on G. Henning and Hedetniemi (2003) showed that the problem of computing gamma r(G) is NP-Hard, even when restricted to bipartite or chordal graphs. This suggests finding gamma r(G) for special classes of graphs or obtaining good bounds on this invariant. In this article, we obtain closed formulae and tight bounds for the weak Roman domination number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product. (C) 2018 Elsevier B.V. All rights reserved.