On Algorithmic Enumeration of Higher-Order Delaunay Triangulations

In the pursuit of realistic terrain models, Gudmundsson, Hammar, and van Kreveld introduced higher-order Delaunay triangulations. A usual Delaunay triangulation is a 0order Delaunay triangulation, thus unique for a non-degenerate point set, while order-k Delaunay triangulations can be non-unique when k ≥ 1. In this work, we propose an algorithm to list all order-k Delaunay triangulations of a given non-degenerate point set on the plane, when k ≤ 2, in polynomial time per triangulation. The main technique is the reverse search due to Avis and Fukuda, which exploits the connectedness of a certain graph over all objects to be listed. We also show that the same technique is unlikely to work for k ≥ 3 by exhibiting an example on which the associated graph is disconnected.