The 2-Domination and 2-Bondage Numbers of Grid Graphs

TL;DR

This paper determines the 2-domination and 2-bondage numbers for grid graphs with dimensions 2 to 4, providing exact values and advancing understanding of domination parameters in grid structures.

Contribution

It explicitly calculates the 2-domination and 2-bondage numbers for small grid graphs, filling a gap in the literature on domination parameters in these graphs.

Findings

Exact values of $2$-domination numbers for $G_{m,n}$ with $2 \u2264 m \u2264 4$.

Exact values of 2-bondage numbers for these grid graphs.

Provides formulas and methods for calculating domination parameters in small grid graphs.

Abstract

Let $p$ be a positive integer and $G=(V,E)$ be a simple graph. A subset $D\subseteq V$ is a $p$-dominating set if each vertex not in $D$ has at least $p$ neighbors in $D$. The $p$-domination number $\g_p(G)$ is the minimum cardinality among all $p$-dominating sets of $G$. The $p$-bondage number $b_p(G)$ is the cardinality of a smallest set of edges whose removal from $G$ results in a graph with a $p$-domination number greater than the $p$-domination number of $G$. In this note we determine the 2-domination number $\g_2$ and 2-bondage number $b_2$ for the grid graphs $G_{m,n}=P_m\times P_n$ for $2\leq m\leq 4$.