Counting Triangulations of Configurations

Calculating the number of Euclidean triangulations of a convex polygon P with vertices in a finite subset C ⊂ R2 containing all vertices of P seems to be difficult and has attracted some interest, both from an algorithmic and a theoretical point of view, see for instance [1], [2], [3], [4], [5], [7], [9], [10], [11]. The aim of this paper is to describe a class of configurations, convex near-gons, for which this problem can sometimes be solved in a satisfactory way. Loosely speaking, a convex near-gon is an infinitesimal perturbation of a weighted convex polygon, a convex polygon with edges subdivided by additional points according to weights. Our main result shows that the triangulation polynomial enumerating all triangulations of a convex nearpolygon is defined in a straightforward way in terms of edge-polynomials associated to the “perturbed” edges of the convex near-gone. These polynomials are difficult to compute in general except in a few special cases. We present a few algorithms related to them. One of these algorithms yields also a general purpose algorithm (unfortunately of exponential complexity), for computing arbitrary triangulation polynomials. This algorithm, based on a transfer matrix, is fairly simple and it would surely be interesting to compare its performance with existing algorithms, like for instance the algorithm described in [1].