Improved Bounds for Guarding Plane Graphs with Edges

TL;DR

This paper improves upper bounds on the number of edges needed to guard any planar graph, providing simpler proofs and tighter bounds that depend on face structures, advancing understanding of edge guarding in plane graphs.

Contribution

The paper offers new, simplified proofs for existing bounds and introduces tighter bounds for edge guarding in plane graphs, including bounds based on quadrilateral faces.

Findings

Guarding with at most 2n/5 edges via inductive proof.

Improved bound of 3n/8 edges for general plane graphs.

Bound of n/3 + α/9 edges depending on quadrilateral faces.

Abstract

An "edge guard set" of a plane graph $G$ is a subset $\Gamma$ of edges of $G$ such that each face of $G$ is incident to an endpoint of an edge in $\Gamma$. Such a set is said to guard $G$. We improve the known upper bounds on the number of edges required to guard any $n$-vertex embedded planar graph $G$: 1- We present a simple inductive proof for a theorem of Everett and Rivera-Campo (1997) that $G$ can be guarded with at most $ \frac{2n}{5}$ edges, then extend this approach with a deeper analysis to yield an improved bound of $\frac{3n}{8}$ edges for any plane graph. 2- We prove that there exists an edge guard set of $G$ with at most $\frac{n}{3}+\frac{\alpha}{9}$ edges, where $\alpha$ is the number of quadrilateral faces in $G$. This improves the previous bound of $\frac{n}{3} + \alpha$ by Bose, Kirkpatrick, and Li (2003). Moreover, if there is no short path between any two quadrilateral faces in $G$, we show that $\frac{n}{3}$ edges suffice, removing the dependence on $\alpha$.