On the disjunctive domination numbers of the torus grid graphs

TL;DR

This paper investigates the disjunctive domination numbers in torus grid graphs, providing bounds and exact values for specific cases, advancing understanding of domination parameters in these structures.

Contribution

It offers new bounds and exact calculations for disjunctive domination numbers in torus grid graphs, including specific cases like C_3×C_n and C_4×C_n.

Findings

Bounds for disjunctive domination numbers in torus grid graphs

Exact disjunctive domination numbers for C_3×C_n and C_4×C_n

Enhanced understanding of domination parameters in grid graphs

Abstract

Let $\Gamma=(V,E)$ be a graph. A disjunctive dominating set in $\Gamma$ is a set $S\subseteq V$ such that every vertex not in $S$ is adjacent to a vertex of $S$, or has at least two vertices in $S$ at distance $2$ from it. The minimum cardinality of a disjunctive dominating set in $\Gamma$ is called the disjunctive domination number of $\Gamma$. In this paper, we give bounds for the disjunctive domination numbers of the torus grid graphs $C_m\Box C_n$, and determine the disjunctive domination numbers of $C_3\Box C_n$ and $C_4\Box C_n$.