Double domination in lexicographic product graphs

TL;DR

This paper investigates the double domination number in lexicographic product graphs, providing tight bounds and formulas based on the properties of the factor graphs, advancing understanding of domination concepts in complex graph structures.

Contribution

It introduces new bounds and closed formulas for the double domination number in lexicographic product graphs, linking it to invariants of the component graphs.

Findings

Derived tight bounds for double domination number.

Established closed formulas for specific graph classes.

Connected double domination to properties of factor graphs.

Abstract

In a graph $G$, a vertex dominates itself and its neighbours. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The minimum cardinality among all double dominating sets of $G$ is the double domination number. In this article, we obtain tight bounds and closed formulas for the double domination number of lexicographic product graphs $G\circ H$ in terms of invariants of the factor graphs $G$ and $H$.