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

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