Linear transformation distance for bichromatic matchings

TL;DR

This paper proves that the transformation graph of bichromatic matchings on a point set is always connected and has a diameter at most 2n, providing an alternative proof and tight bound for this geometric problem.

Contribution

It offers an alternative proof of the connectivity of the transformation graph and establishes a tight upper bound of 2n for its diameter.

Findings

Transformation graph is always connected.

Diameter of the transformation graph is at most 2n.

The bound of 2n is asymptotically tight.

Abstract

Let $P=B\cup R$ be a set of $2n$ points in general position, where $B$ is a set of $n$ blue points and $R$ a set of $n$ red points. A \emph{$BR$-matching} is a plane geometric perfect matching on $P$ such that each edge has one red endpoint and one blue endpoint. Two $BR$-matchings are compatible if their union is also plane. The \emph{transformation graph of $BR$-matchings} contains one node for each $BR$-matching and an edge joining two such nodes if and only if the corresponding two $BR$-matchings are compatible. In SoCG 2013 it has been shown by Aloupis, Barba, Langerman, and Souvaine that this transformation graph is always connected, but its diameter remained an open question. In this paper we provide an alternative proof for the connectivity of the transformation graph and prove an upper bound of $2n$ for its diameter, which is asymptotically tight.