Dominating the direct product of two graphs through total Roman strategies

TL;DR

This paper investigates the total Roman domination number of the direct product of two graphs, establishing bounds, characterizations, and exact values for specific classes, advancing understanding of domination parameters in graph products.

Contribution

It provides new bounds, characterizations, and exact values for the total Roman domination number of direct product graphs, a novel focus in graph domination theory.

Findings

Established bounds relating $\, ext{γ}_{tR}(G imes H)$ to classical parameters.

Characterized graphs with small total Roman domination numbers in their direct products.

Calculated exact values for specific classes of direct product graphs.

Abstract

Given a graph $G$ without isolated vertices, a total Roman dominating function for $G$ is a function $f : V(G)\rightarrow \{0,1,2\}$ such that every vertex with label 0 is adjacent to a vertex with label 2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number $\gamma_{tR}(G)$ of $G$ is the smallest possible value of $\sum_{v\in V(G)}f(v)$ among all total Roman dominating functions $f$. The total Roman domination number of the direct product $G\times H$ of the graphs $G$ and $H$ is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between $\gamma_{tR}(G\times H)$ and some classical domination parameters for the factors are given. Characterizations of the direct product graphs $G\times H$ achieving small values ($\le 7$) for $\gamma_{tR}(G\times H)$ are presented, and exact values for $\gamma_{tR}(G\times H)$ are deduced, while considering various specific direct product classes.