Secure Total Domination Number in Maximal Outerplanar Graphs
<|reference_start|>Secure Total Domination Number in Maximal Outerplanar Graphs: 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.<|reference_end|>
@article{aita2024secure,
    title={Secure Total Domination Number in Maximal Outerplanar Graphs},
    author={Yasufumi Aita and Toru Araki},
    journal={arXiv preprint arXiv:2403.03404},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.03404},
    primaryClass={math.CO cs.DM}
}