Minimum number of partial triangulations

TL;DR

This paper establishes a lower bound on the number of partial triangulations of n points in the plane, showing it equals the (n-2)-nd Catalan number, with a complete characterization of cases where equality holds.

Contribution

It proves a tight lower bound on the count of partial triangulations for any point set, linking it to Catalan numbers and characterizing equality cases.

Findings

Number of partial triangulations ≥ Catalan number C_{n-2}

Equality cases characterized for convex n-gons

Result is tight for convex polygons

Abstract

We show that the number of partial triangulations of a set of $n$ points on the plane is at least the $(n-2)$-nd Catalan number. This is tight for convex $n$-gons. We also describe all the equality cases.