On the Roman domination in the lexicographic product of graphs

A Roman dominating function of a graph G = (V, E) is a function f : V -> {0, 1, 2} such that every vertex with f (v) = 0 is adjacent to some vertex with f(v) = 2. The Roman domination number of G is the minimum of w(f) = Sigma(nu is an element of V)f(v) over all such functions. Using a new concept of the so-called dominating couple we establish the Roman domination number of the lexicographic product of graphs. We also characterize Roman graphs among the lexicographic product of graphs. (C) 2012 Elsevier B.V. All rights reserved.