  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.
