Happy endings for flip graphs

TL;DR

This paper demonstrates that triangulations of certain point sets can be embedded into hypercubes, enabling efficient computation of flip distances, with the key condition being the absence of empty convex pentagons.

Contribution

It characterizes when flip graphs of point sets can be isometrically embedded into hypercubes, linking geometric properties to combinatorial structures.

Findings

Flip graphs of specific point sets embed into hypercubes

Efficient computation of flip distances for these sets

Characterization based on absence of empty convex pentagons

Abstract

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.