Roman domination in direct product graphs and rooted product graphs

TL;DR

This paper investigates the Roman domination number in direct and rooted product graphs, providing bounds and characterizations based on various graph parameters, advancing understanding of domination concepts in complex graph structures.

Contribution

It offers new bounds for Roman domination in direct product graphs and characterizes possible values for rooted product graphs, linking these to key graph parameters.

Findings

Derived tight bounds for Roman domination in direct product graphs.

Identified three possible values for Roman domination in rooted product graphs.

Provided characterizations for rooted product graphs achieving each value.

Abstract

Let $G$ be a graph with vertex set $V(G)$. A function $f:V(G)\rightarrow \{0,1,2\}$ is a Roman dominating function on $G$ if every vertex $v\in V(G)$ for which $f(v)=0$ is adjacent to at least one vertex $u\in V(G)$ such that $f(u)=2$. The Roman domination number of $G$ is the minimum weight $\omega(f)=\sum_{x\in V(G)}f(x)$ among all Roman dominating functions $f$ on $G$. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving some parameters of the factors, which include the domination, (total) Roman domination, and packing numbers among others. On the other hand, we prove that the Roman domination number of rooted product graphs can attain only three possible values, which depend on the order, the domination, and the Roman domination numbers of the factors in the product. In addition, theoretical characterizations of the classes of rooted product graphs achieving each of these three possible values are given.