Detecting weakly simple polygons

Abstract

A closed curve in the plane is weakly simple if it is the limit (in the Frechet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O (n log n) time, improving an earlier O (n3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O (n2log n) time. We also describe algorithms that detect weak simplicity in 0 (n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.