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On the Roman domination in the lexicographic product of graphs
Kraner Šumenjak, Tadeja ; Repolusk, Polona ; Tepeh, Aleksandra
A Roman dominating function of a graph ▫$G = (V,E)$▫ is a function ▫$f \colon V \to \{0,1,2\}$▫ 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 ▫$w(f) = \sum_{v \in V}f(v)$▫ 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.
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