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.
翻译：本文的目的是为了获得完美支配数字的封闭公式、罗马统治数字和完美罗马统治数字的词汇产品图表。我们表明,就前两个参数而言,这些公式可以比较容易地获得。当它涉及完美的罗马统治数字时,情况则大不相同。在这种情况下,我们获得一般界限,然后为达到界限提供足够和(或)必要的条件。我们还讨论完美罗马图的例子,并描述完美罗马统治数字等于罗马统治数字的词汇产品图。
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