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arXiv · 2506.07000

The $k$-Total Bondage Number of a Graph

Jean-Pierre Appel·Gabby Fischberg·Kyle Kelley·Nathan Shank·Eliel Sosis
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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.

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Jean-Pierre Appel, Gabby Fischberg, Kyle Kelley, Nathan Shank, Eliel Sosis. 2025-06-08. The $k$-Total Bondage Number of a Graph. https://arxiv.org/abs/2506.07000

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