Lattice of Triangulations: the proof and an algorithm

TL;DR

This paper proves that the set of polygon triangulations forms a lattice under a specific order and introduces an efficient algorithm to find the infimum of two triangulations, extending previous work on flip graphs.

Contribution

It establishes the lattice structure of triangulations and presents a new, efficient algorithm for computing their infimum, building on prior flip graph properties.

Findings

Triangulations form a lattice under a defined order.

The proposed algorithm efficiently finds the infimum of two triangulations.

The lattice property is rigorously proven based on existing flip graph theorems.

Abstract

In this paper, we prove that the set of triangulations of a polygon can be equipped with an order to become a lattice. First, we define this order. In [HN99], authors defined the flip operator and then prove some properties of the graph of triangulations. We use their theorems and extend them to construct the lattice of triangulations. We prove this lattice property and introduce an elegant algorithm which correctness is induced from the proof. The complexity of this algorithm will be considered. This algorithm is efficient to find the infimum of a pair of triangulations.