
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.
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