Maximal Independent Sets in Planar Triangulations

TL;DR

This paper proves that every planar triangulation on n vertices has a maximal independent set of size at most n/3, confirming a recent conjecture and strengthening known domination results in triangulations.

Contribution

It establishes an upper bound of n/3 for the size of maximal independent sets in all planar triangulations, confirming a conjecture and advancing structural understanding.

Findings

Maximal independent set size is at most n/3 in planar triangulations

Confirms a conjecture by Botler, Fernandes, and Gutiérrez

Strengthens the known bounds on dominating sets in triangulations

Abstract

We show that every planar triangulation on $n$ vertices has a maximal independent set of size at most $n/3$. This affirms a conjecture by Botler, Fernandes and Guti\'errez [Electron.\ J.\ Comb., 2024], which in turn would follow if an open question of Goddard and Henning [Appl.\ Math.\ Comput., 2020] which asks if every planar triangulation has three disjoint maximal independent sets were answered in the affirmative. Since a maximal independent set is a special type of dominating set (independent dominating set), this is a structural strengthening of a major result by Matheson and Tarjan [Eur.\ J.\ Comb., 1996] that every triangulated disc has a dominating set of size at most $n/3$, but restricted to triangulations.