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

TL;DR

This paper derives formulas and bounds for various domination parameters in lexicographic product graphs, highlighting differences in complexity among them and characterizing special cases such as perfect Roman graphs.

Contribution

It provides new formulas and bounds for domination numbers in lexicographic product graphs, especially for perfect Roman domination, and characterizes cases where these parameters coincide.

Findings

Formulas for perfect and Roman domination numbers in lexicographic products.

Bounds and conditions for perfect Roman domination number.

Characterization of perfect Roman graphs.

Abstract

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.