Dominating Plane Triangulations

TL;DR

This paper improves bounds on the domination number of certain plane triangulations, proving it is at most approximately 31% of the vertices for large graphs, advancing understanding of domination in planar graphs.

Contribution

It establishes new upper bounds on the domination number for Hamiltonian and 4-connected plane triangulations with minimum degree at least 4.

Findings

For Hamiltonian plane triangulations with minimum degree ≥ 4, domination number ≤ max{2n/7, 5n/16}.

For 4-connected plane triangulations with n ≥ 26, domination number ≤ 5n/16.

Progress towards the conjecture that the domination number can be bounded by n/4 for large n.

Abstract

In 1996, Tarjan and Matheson proved that if $G$ is a plane triangulated disc with $n$ vertices, $\gamma (G)\le n/3$, where $\gamma (G)$ denotes the domination number of $G$. Furthermore, they conjectured that the constant $1/3$ could be improved to $1/4$ for sufficiently large $n$. Their conjecture remains unsettled. In the present paper, it is proved that if $G$ is a hamiltonian plane triangulation with $|V(G)|=n$ vertices and minimum degree at least 4, then $\gamma (G)\le\max\{\lceil 2n/7\rceil, \lfloor 5n/16\rfloor\}$. It follows immediately that if $G$ is a 4-connected plane triangulation with $n$ vertices, then $\gamma (G)\le\max\{\lceil 2n/7\rceil, \lfloor 5n/16\rfloor\} $. It then follows that if $n\ge 26$, then $\gamma (G)\le \lfloor 5n/16\rfloor$.