Flips in Odd Matchings

TL;DR

This paper studies the flip graph of matchings on an odd set of points in the plane, proving its connectivity, providing an upper bound on its diameter, and establishing a tight bound for convex position points.

Contribution

It introduces the graph of plane matchings on odd point sets, proves its connectivity, and bounds its diameter, including a tight bound for convex configurations.

Findings

The flip graph $GM_ ext{P}$ is connected.

An $O(n^2)$ upper bound on the diameter of $GM_ ext{P}.

A tight $ heta(n)$ bound for convex position points.

Abstract

Let $\mathcal{P}$ be a set of $n=2m+1$ points in the plane in general position. We define the graph $GM_\mathcal{P}$ whose vertex set is the set of all plane matchings on $\mathcal{P}$ with exactly $m$ edges. Two vertices in $GM_\mathcal{P}$ are connected if the two corresponding matchings have $m-1$ edges in common. In this work we show that $GM_\mathcal{P}$ is connected and give an upper bound of $O(n^2)$ on its diameter. Moreover, we present a tight bound of $\Theta(n)$ for the diameter of the flip graph of points in convex position.