Triangulations of nearly convex polygons

Counting Euclidean triangulations with vertices in a finite set $\C$ of the convex hull $\conv(\C)$ of $\C$ is difficult in general, both algorithmically and theoretically. The aim of this paper is to describe nearly convex polygons, a class of configurations for which this problem can be solved to some extent. Loosely speaking, a nearly convex polygon is an infinitesimal perturbation of a weakly convex polygon (a convex polygon with edges subdivided by additional points). Our main result shows that the triangulation polynomial, enumerating all triangulations of a nearly convex polygon, is defined in a straightforward way in terms of polynomials associated to the ``perturbed'' edges.