Flips in Higher Order Delaunay Triangulations.

We study the flip graph of higher order Delaunay triangulations. A triangulation of a set S of n points in the plane is order- k Delaunay if for every triangle its circumcircle encloses at most k points of S . The flip graph of S has one vertex for each possible triangulation of S , and an edge connecting two vertices when the two corresponding triangulations can be transformed into each other by a flip (i.e., exchanging the diagonal of a convex quadrilateral by the other one). The flip graph is an essential structure in the study of triangulations, but until now it had been barely studied for order- k Delaunay triangulations. In this work we show that, even though the order- k flip graph might be disconnected for k ≥ 3 , any order- k triangulation can be transformed into some other order- k triangulation by at most k - 1 flips, such that the intermediate triangulations are of order at most 2 k - 2 , in the following settings: (1) for any k ≥ 0 when S is in convex position, and (2) for any k ≤ 5 and any point set S . Our results imply that the flip distance between two order- k triangulations is O ( kn ), as well as an efficient enumeration algorithm.