Disjunctive domination in maximal outerplanar graphs

Abstract

A disjunctive dominating set of a graph $G$ is a set $D \subseteq V(G)$ such that every vertex in $V(G)\setminus D$ has a neighbor in $D$ or has at least two vertices in $D$ at distance $2$ from it. The disjunctive domination number of $G$, denoted by $\gamma_2^d(G)$, is the minimum cardinality of a disjunctive dominating set of $G$. In this paper, we show that if $G$ is a maximal outerplanar graph of order $n \ge 7$ with $k$ vertices of degree $2$, then $\gamma_2^d(G)\le \lfloor\frac{2}{9}(n+k)\rfloor$, and this bound is sharp.