Secure Total Domination Number in Maximal Outerplanar Graphs
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2024
- DOI:
- 10.48550/arXiv.2403.03404
- arXiv:
- arXiv:2403.03404
- Bibcode:
- 2024arXiv240303404A
- Keywords:
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- Mathematics - Combinatorics;
- Computer Science - Discrete Mathematics;
- 05C69;
- 05C10;
- G.2.2