A Bound on the Edge-Flipping Distance between Triangulations (Revisiting the Proof)

TL;DR

This paper provides a comprehensive and detailed proof of a fundamental bound on the flip distance between planar triangulations, relating it to the number of edge intersections, and clarifies previous gaps in the literature.

Contribution

It offers a complete, case-based proof of the upper bound on flip distance, generalizing previous results and addressing gaps in earlier proofs.

Findings

Established a detailed proof of the flip distance bound

Generalized the result to a broader setting

Clarified previous gaps in the proof literature

Abstract

We revisit here a fundamental result on planar triangulations, namely that the flip distance between two triangulations is upper-bounded by the number of proper intersections between their straight-segment edges. We provide a complete and detailed proof of this result in a slightly generalised setting using a case-based analysis that fills several gaps left by previous proofs of the result.