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arXiv · 2403.03404

Secure Total Domination Number in Maximal Outerplanar Graphs

Yasufumi Aita·Toru Araki
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Abstract

A subset $S$ of vertices in a graph $G$ is a secure total dominating set of $G$ if $S$ is a total dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup \{u\}$ is also a total dominating set of $G$. We show that if $G$ is a maximal outerplanar graph of order $n$, then $G$ has a total secure dominating set of size at most $\lfloor 2n/3 \rfloor$. Moreover, if an outerplanar graph $G$ of order $n$, then each secure total dominating set has at least $\lceil (n+2)/3 \rceil$ vertices. We show that these bounds are best possible.

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Yasufumi Aita, Toru Araki. 2024-04-09. Secure Total Domination Number in Maximal Outerplanar Graphs. https://arxiv.org/abs/2403.03404

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