On the Roman domination in the lexicographic product of graphs

A Roman dominating function of a graph G=(V,E)G=(V,E) is a function f:V→{0,1,2}f:V→{0,1,2} such that every vertex with f(v)=0f(v)=0 is adjacent to some vertex with f(v)=2f(v)=2. The Roman domination number of GG is the minimum of w(f)=∑v∈Vf(v)w(f)=∑v∈Vf(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.