On Numbers of Pseudo-Triangulations

TL;DR

This paper establishes exponential bounds on the maximum number of pseudo-triangulations and pointed pseudo-triangulations that can be embedded over a fixed set of points or contained within a specific triangulation, advancing understanding of their combinatorial complexity.

Contribution

It provides new exponential bounds on the counts of pseudo-triangulations and pointed pseudo-triangulations, both in fixed point sets and within specific triangulations, which were previously unknown.

Findings

Maximum pointed pseudo-triangulations in a triangulation: O(5.45^N) upper bound, Ω(2.41^N) lower bound.

Number of all pseudo-triangulations in a triangulation: O*(6.54^N) upper bound, Ω(3.30^N) lower bound.

Pointed pseudo-triangulations over any point set: at most 89.1^N, general pseudo-triangulations: at most 120^N.

Abstract

We study the maximum numbers of pseudo-triangulations and pointed pseudo-triangulations that can be embedded over a specific set of points in the plane or contained in a specific triangulation. We derive the bounds $O(5.45^N)$ and $\Omega (2.41^N)$ for the maximum number of pointed pseudo-triangulations that can be contained in a specific triangulation over a set of $N$ points. For the number of all pseudo-triangulations contained in a triangulation we derive the bounds $O^*(6.54^N)$ and $\Omega (3.30^N)$. We also prove that $O^*(89.1^N)$ pointed pseudo-triangulations can be embedded over any specific set of $N$ points in the plane, and at most $120^N$ general pseudo-triangulations.