Independent dominating sets in planar triangulations

Abstract

In 1996, Matheson and Tarjan proved that every planar triangulation on \(n\) vertices contains a dominating set %, i.e., a set \(S\) that contains a neighbor of every vertex not in \(S\), of size at most \(n/3\), and conjectured that this upper bound can be reduced to \(n/4\) when $n$ is sufficiently large. In this paper, we consider the analogous problem for independent dominating sets: What is the minimum \(\eps\) for which every planar triangulation on \(n\) vertices contains an independent dominating set of size at most \(\eps n\)? We prove that \(2/7 \leq \eps \leq 3/8\).