Disjoint Total Dominating Sets in Near-Triangulations

TL;DR

This paper proves that all simple planar near-triangulations with minimum degree three have two disjoint total dominating sets, confirming a conjecture and extending known results in graph theory.

Contribution

It establishes that such near-triangulations always contain two disjoint total dominating sets, broadening the class of graphs known to have this property.

Findings

Every simple planar near-triangulation with minimum degree at least three contains two disjoint total dominating sets.

Includes all simple planar triangulations except the triangle.

Confirms a conjecture by Goddard and Henning.

Abstract

We show that every simple planar near-triangulation with minimum degree at least three contains two disjoint total dominating sets. The class includes all simple planar triangulations other than the triangle. This affirms a conjecture of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88 (2018) 174-191].