Flips in Edge-Labelled Pseudo-Triangulations

TL;DR

This paper demonstrates that a quadratic number of flips suffices to transform any edge-labelled pointed pseudo-triangulation into another, and provides bounds for transformations involving insertion, deletion, and exchanging flips based on geometric parameters.

Contribution

It establishes bounds on the number of flips needed to transform between edge-labelled pseudo-triangulations, extending understanding of flip operations in geometric graph transformations.

Findings

O(n^2) flips suffice for any edge-labelled pointed pseudo-triangulation transformation

Transformations involving insertion, deletion, and exchanging flips are bounded by O(n log c + h log h) flips

The bounds depend on convex layers and hull points of the point set

Abstract

We show that $O(n^2)$ exchanging flips suffice to transform any edge-labelled pointed pseudo-triangulation into any other with the same set of labels. By using insertion, deletion and exchanging flips, we can transform any edge-labelled pseudo-triangulation into any other with $O(n \log c + h \log h)$ flips, where $c$ is the number of convex layers and $h$ is the number of points on the convex hull.