Improved Bounds for Guarding Plane Graphs with Edges

n edge guard set of a plane graph G is a subset of edges of G such that each face of G is incident to an endpoint of an edge in . 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 (Comput Geom Theory Appl 7:201–203, 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 + α/9 edges, where α is the number of quadrilateral faces in G . This improves the previous bound of n/3 + α 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 n/3 edges suffice, removing the dependence on α .