Flipping edges in triangulations

Abstract

In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least [(n - 4)/2] edges that can be flipped. We also prove that O(n + k2) flips are sufficient to transform any triangulation of an n-gon with k reflex vertices into any other triangulation. We produce examples of n-gons with triangulations T and T′ such that to transform T into T′ requires Ω(n2) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips.