Double domination in maximal outerplanar graphs

TL;DR

This paper establishes tight bounds on the double domination number in maximal outerplanar graphs, providing new theoretical limits and analyzing special cases like striped graphs.

Contribution

It introduces new upper bounds for the double domination number in maximal outerplanar graphs, including cases with vertices of degree two, and proves their tightness.

Findings

Bound of    for  graphs.

Improved bounds involving the number of degree-2 vertices.

Analysis of striped 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 $n\geq 3$ vertices, then $\gamma_{\times 2}(G)\leq \lfloor \frac{2n}{3}\rfloor$. Further, if $n\geq 4$, then $\gamma_{\times 2}(G)\leq \min \{\lfloor \frac{n+t}{2}\rfloor, n-t\}$, 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.