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arXiv · 2308.02754

Independent dominating sets in planar triangulations

Fábio Botler·Cristina G. Fernandes·Juan Gutiérrez
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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.

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Fábio Botler, Cristina G. Fernandes, Juan Gutiérrez. 2023-08-05. Independent dominating sets in planar triangulations. https://arxiv.org/abs/2308.02754

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