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)[email protected]?"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.

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