Dominating maximal outerplane graphs and Hamiltonian plane triangulations

TL;DR

This paper investigates bounds on the domination number of certain planar graphs, providing new upper bounds for maximal outerplane graphs and Hamiltonian plane triangulations, and correcting previous inaccuracies in the literature.

Contribution

It establishes new upper bounds on the domination number for maximal outerplane graphs and Hamiltonian plane triangulations, and corrects prior published results.

Findings

Maximal outerplane graphs have a domination number at most rac{n+k}{4}",

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Abstract

Let $G$ be a graph and $\gamma (G)$ denote the domination number of $G$, i.e. the cardinality of a smallest set of vertices $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. Matheson and Tarjan conjectured that a plane triangulation with a sufficiently large number $n$ of vertices has $\gamma(G)\le n/4$. Their conjecture remains unsettled. In the present paper, we show that: (1) a maximal outerplane graph with $n$ vertices has $\gamma(G)\le \lceil \frac{n+k} 4\rceil$ where $k$ is the number of pairs of consecutive degree 2 vertices separated by distance at least 3 on the boundary of $G$; and (2) a Hamiltonian plane triangulation $G$ with $n \ge 23$ vertices has $\gamma (G)\le 5n/16 $. We also point out and provide counterexamples for several recent published results of Li et al in [Discrete Appl. Math.198 (2016) 164-169] on this topic which are incorrect.