Perfect domination, Roman domination and perfect Roman domination in lexicographic product graphs

The aim of this paper is to obtain closed formulas for the perfect domination number, the Roman domination number and the perfect Roman domination number of lexicographic product graphs. We show that these formulas can be obtained relatively easily for the case of the first two parameters. The picture is quite different when it concerns the perfect Roman domination number. In this case, we obtain general bounds and then we give sufficient and/or necessary conditions for the bounds to be achieved. We also discuss the case of perfect Roman graphs and we characterize the lexicographic product graphs where the perfect Roman domination number equals the Roman domination number.