Double domination in maximal outerplanar graphs

Abstract

In a graph $G$, a vertex dominates itself and its neighbors. A subset $S\subseteq V(G)$ is said to be a double dominating set of $G$ if $S$ dominates every vertex of $G$ at least twice. The double domination number ${\gamma }_{\times 2}(G)$ is the minimum cardinality of a double dominating set of $G$. We show that if $G$ is a maximal outerplanar graph on $n\ge 3$ vertices, then ${\gamma }_{\times 2}(G)\le \lfloor \frac{2n}{3}\rfloor$. Further, if $n\ge 4$, then ${\gamma }_{\times 2}(G)\le \min \{\lfloor \frac{n+t}{2}\rfloor ,n-t\}$, where $t$ is the number of vertices of degree $2$ in $G$. These bounds are shown to be tight. In addition, we also study the case that $G$ is a striped maximal outerplanar graph.