The domination number of plane triangulations

TL;DR

This paper introduces weak near-triangulations and proves that every plane triangulation with more than six vertices has a dominating set of size at most approximately 0.32 times the number of vertices, improving previous bounds.

Contribution

The paper defines weak near-triangulations and establishes a tighter upper bound on the domination number of plane triangulations, advancing graph theory knowledge.

Findings

Every plane triangulation on n > 6 vertices has a dominating set of size at most 17n/53.

The class of weak near-triangulations is closed under certain graph operations.

Improves the previous bound of n/3 for domination number in plane triangulations.

Abstract

We introduce a class of plane graphs called weak near-triangulations, and prove that this class is closed under certain graph operations. Then we use the properties of weak near-triangulations to prove that every plane triangulation on $n>6$ vertices has a dominating set of size at most $17n/53$. This improves the bound $n/3$ obtained by Matheson and Tarjan.