Improved Bounds for Guarding Plane Graphs with Edges

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 2n/5 edges, then extend this approach with a deeper analysis to yield an improved bound of 3n/8 edges for any plane graph. 2) We prove that there exists an edge guard set of G with at most n/(3) + alpha/9 edges, where alpha is the number of quadrilateral faces in G. This improves the previous bound of 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 n/(3) edges suffice, removing the dependence on alpha.