Improved Bounds for Guarding Plane Graphs with Edges

Abstract

An edge guard set of a plane graph G is a subset $\varGamma$ of edges of G such that each face of G is incident to an endpoint of an edge in $\varGamma$. Such a set is said to guardG. 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 (Comput Geom Theory Appl 7:201–203, 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 et al. (Comput Geom Theory Appl 26(3):209–219, 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 $\alpha$.