A Lower bound for Secure Domination Number of an Outerplanar Graph
<|reference_start|>A Lower bound for Secure Domination Number of an Outerplanar Graph: A subset $S$ of vertices in a graph $G$ is a secure dominating set of $G$ if $S$ is a 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 dominating set of $G$. The secure domination number of $G$, denoted by $\gamma_{s}(G)$, is the cardinality of a smallest secure dominating sets of $G$. In this paper, we prove that for any outerplanar graph with $n \geq 4$ vertices, $\gamma_{s}(G) \geq (n+4)/5$ and the bound is tight.<|reference_end|>
@article{araki2024a,
    title={A Lower bound for Secure Domination Number of an Outerplanar Graph},
    author={Toru Araki},
    journal={arXiv preprint arXiv:2403.06493},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.06493},
    primaryClass={math.CO cs.DM}
}