Guarding Quadrangulations and Stacked Triangulations with Edges

TL;DR

This paper studies the minimal number of edges needed to guard all faces in specific classes of planar graphs, providing tight bounds for quadrangulations and stacked triangulations.

Contribution

It establishes tight bounds on the number of edges required to guard all faces in quadrangulations and stacked triangulations, improving previous results.

Findings

At most  n/3 edges guard all faces in quadrangulations.

 (n-2)/4 edges are necessary for some quadrangulations.

 2n/7 edges suffice for stacked triangulations, and this bound is nearly optimal.

Abstract

Let $G = (V,E)$ be a plane graph. A face $f$ of $G$ is guarded by an edge $vw \in E$ if at least one vertex from $\{v,w\}$ is on the boundary of $f$. For a planar graph class $\mathcal{G}$ we ask for the minimal number of edges needed to guard all faces of any $n$-vertex graph in $\mathcal{G}$. We prove that $\lfloor n/3 \rfloor$ edges are always sufficient for quadrangulations and give a construction where $\lfloor (n-2)/4 \rfloor$ edges are necessary. For $2$-degenerate quadrangulations we improve this to a tight upper bound of $\lfloor n/4 \rfloor$ edges. We further prove that $\lfloor 2n/7 \rfloor$ edges are always sufficient for stacked triangulations (that are the $3$-degenerate triangulations) and show that this is best possible up to a small additive constant.