Flipping Edges in Triangulations of Point Sets, Polygons and Maximal Planar Graphs

A triangulation of a point set Pn is a partitioning of the convex hull Conv (Pn) into a set of triangles with disjoint interiors such that the vertices of these triangles are in Pn, and no element of Pn lies in the interior of any of these triangles. An edge e of a triangulation T is called flippable if it is contained in the boundary of two triangles of T , and the union of these triangles forms a convex quadrilateral C. By flipping e we mean the operation of deleting e from T and replacing it by the other diagonal of C. A triangulation of a polygon Qn is a partition of Qn into a set of n−2 triangles with disjoint interiors such that the edges of these triangles are vertices of Qn. In this paper we will prove that any triangulation of a point set (polygon) can be transformed into any other by a sequence of flips. We will also prove that there are triangulations of point sets (polygons) such that to transform one into the other takes O(n2) flips. We prove that any triangulation of any set of points contains at least ⌊ 2 ⌋ flippable edges. Motivated by this result, we generalize the concept of flipping edges to that of simultaneously flipping sets of independent ∗Supported by NSERC of Canada. The results presented here are taken from three papers co-authored by the author. Sections 2, 3 and 4 are taken from papers co-authored with F. Hurtado and M. Noy [17, 18]. Section 5 is taken from a joint paper co-authored with Z. Gao and J. Wang [9].