Independent dominating sets in planar triangulations

TL;DR

This paper investigates bounds on the size of independent dominating sets in planar triangulations, establishing new upper and lower bounds and improving existing bounds for specific subclasses.

Contribution

It provides new bounds for the minimum proportion of vertices in independent dominating sets in near planar triangulations and specific subclasses.

Findings

Lower bound of 2/7 for independent dominating sets

Upper bound of 5/12 for near planar triangulations

Improved upper bounds to 3/8 and 1/3 for specific cases

Abstract

In 1996, Matheson and Tarjan proved that every near planar triangulation on $n$ vertices contains a dominating set of size at most $n/3$, and conjectured that this upper bound can be reduced to $n/4$ for planar triangulations when $n$ is sufficiently large. In this paper, we consider the analogous problem for independent dominating sets: What is the minimum $\epsilon$ for which every near planar triangulation on $n$ vertices contains an independent dominating set of size at most $\epsilon n$? We prove that $2/7 \leq \epsilon \leq 5/12$. Moreover, this upper bound can be improved to $3/8$ for planar triangulations, and to $1/3$ for planar triangulations with minimum degree 5.