The total bondage number of grid graphs

TL;DR

This paper investigates the total bondage number of grid graphs, providing exact values for certain cases and bounds for others, advancing understanding of graph vulnerability related to domination properties.

Contribution

It determines exact total bondage numbers for (n,2) and (n,3) grid graphs and offers bounds for (n,4), contributing new precise and bounded results in graph theory.

Findings

Exact total bondage numbers for G_{n,2} and G_{n,3}

Upper bounds for G_{n,4}

Enhanced understanding of grid graph vulnerability

Abstract

The total domination number of a graph $G$ without isolated vertices is the minimum number of vertices that dominate all vertices in $G$. The total bondage number $b_t(G)$ of $G$ is the minimum number of edges whose removal enlarges the total domination number. This paper considers grid graphs. An $(n,m)$-grid graph $G_{n,m}$ is defined as the cartesian product of two paths $P_n$ and $P_m$. This paper determines the exact values of $b_t(G_{n,2})$ and $b_t(G_{n,3})$, and establishes some upper bounds of $b_t(G_{n,4})$.