Open guard edges and edge guards in simple polygons

Abstract

An open edge of a simple polygon is the set of points in the relative interior of an edge. We revisit several art gallery problems, previously considered for closed edge guards, using open edge guards. A guard edge of a poly- gon is an edge that sees every point inside the polygon. We show that every simple non-starshaped polygon ad- mits at most one open guard edge, and give a simple new proof that it admits at most three closed guard edges. We characterize open guard edges, and derive an algorithm that finds all open guard edges of a simple n-gon in O(n) time in the RAM model of computation. Finally, we present lower bound constructions for simple polygons with n vertices that require [n/3] open edge guards, and conjecture that this bound is tight.