Triangulations of nearly convex polygons

TL;DR

This paper introduces nearly convex polygons, a class of configurations close to convex polygons, and provides a straightforward polynomial formula to count their triangulations, advancing understanding of triangulation enumeration.

Contribution

It defines nearly convex polygons and derives a simple polynomial expression for counting their triangulations, extending triangulation enumeration methods to this new class.

Findings

Triangulation polynomial for nearly convex polygons is explicitly characterized.

Enumeration of triangulations can be efficiently computed for these polygons.

The approach bridges convex and nearly convex polygon triangulation counting.

Abstract

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.