On dominating sets of maximal outerplanar graphs

Abstract

A dominating set of a graph is a set $S$ of vertices such that every vertex in the graph is either in $S$ or is adjacent to a vertex in $S$. The domination number of a graph $G$, denoted $\gamma \left(G\right)$, is the minimum cardinality of a dominating set of $G$. We show that if $G$ is an $n$-vertex maximal outerplanar graph, then $\gamma \left(G\right)\le (n+t)/4$, where $t$ is the number of vertices of degree $2$ in $G$. We show that this bound is tight for all $t\ge 2$. Upper-bounds for $\gamma \left(G\right)$ are known for a few classes of graphs.