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 $u\in ̸S$, 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 $\lfloor 2n\mathrm{/}3\rfloor$. Moreover, if an outerplanar graph $G$ of order $n$, then each secure total dominating set has at least $\lceil (n+2)\mathrm{/}3\rceil$ vertices. We show that these bounds are best possible.