Total and paired domination numbers of toroidal meshes

TL;DR

This paper calculates the total and paired domination numbers for toroidal meshes, specifically for Cartesian products of cycles, providing exact values for small cases and bounds for larger ones.

Contribution

It determines the total and paired domination numbers of toroidal meshes formed by cycles, extending known results and offering bounds for larger cycle sizes.

Findings

Exact total and paired domination numbers for $C_n \square C_m$ with small $m$

Upper bounds for larger $n, m$ values

Extensions of domination number results to toroidal meshes

Abstract

Let $G$ be a graph without isolated vertices. The total domination number of $G$ is the minimum number of vertices that can dominate all vertices in $G$, and the paired domination number of $G$ is the minimum number of vertices in a dominating set whose induced subgraph contains a perfect matching. This paper determines the total domination number and the paired domination number of the toroidal meshes, i.e., the Cartesian product of two cycles $C_n$ and $C_m$ for any $n\ge 3$ and $m\in\{3,4\}$, and gives some upper bounds for $n, m\ge 5$.