The $k$-Total Bondage Number of a Graph

TL;DR

This paper introduces the concept of the $k$-total bondage number, measuring the minimum edges removal needed to significantly increase a graph's total domination number, and provides exact values for various graph classes.

Contribution

It defines the $k$-total bondage number and derives exact values for specific graph classes, expanding understanding of domination parameters.

Findings

Exact $k$-total bondage values for paths and cycles

Results for wheels, complete, and bipartite graphs

General properties of the $k$-total bondage number

Abstract

Let $G=(V,E)$ be a connected, finite undirected graph. A set $S \subseteq V$ is said to be a total dominating set of $G$ if every vertex in $V$ is adjacent to some vertex in $S$. The total domination number, $\gamma_{t}(G)$, is the minimum cardinality of a total dominating set in $G$. We define the $k$-total bondage of $G$ to be the minimum number of edges to remove from $G$ so that the resulting graph has a total domination number at least $k$ more than $\gamma_{t}(G)$. We establish general properties of $k$-total bondage and find exact values for certain graph classes including paths, cycles, wheels, complete and complete bipartite graphs.