Roman domination in Cartesian product graphs and strong product graphs

TL;DR

This paper investigates the Roman domination number in Cartesian and strong product graphs, exploring how it relates to the domination numbers of the individual factor graphs, thereby advancing understanding of graph domination properties.

Contribution

It introduces new relationships between the Roman domination number of product graphs and the domination numbers of their factors, extending existing graph theory results.

Findings

Established bounds for Roman domination numbers in product graphs

Connected Roman domination numbers with factor graph domination numbers

Provided theoretical insights into domination properties in complex graph products

Abstract

A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex outside of $S$ is adjacent to at least one vertex belonging to $S$. The minimum cardinality of a dominating set for $G$ is called the domination number of $G$. A map $f : V \rightarrow \{0, 1, 2\}$ is a Roman dominating function on a graph $G$ if for every vertex $v$ with $f(v) = 0$, there exists a vertex $u$, adjacent to $v$, such that $f(u) = 2$. The weight of a Roman dominating function is given by $f(V) =\sum_{u\in V}f(u)$. The minimum weight of a Roman dominating function on $G$ is called the Roman domination number of $G$. In this article we study the Roman domination number of Cartesian product graphs and strong product graphs. More precisely, we study the relationships between the Roman domination number of product graphs and the (Roman) domination number of the factors.