Bichromatic compatible matchings

TL;DR

This paper proves that the graph of all bichromatic non-crossing perfect matchings, where edges are compatible if their union remains non-crossing, is connected, resolving an open problem in geometric graph theory.

Contribution

It establishes the connectivity of the compatible bichromatic matching graph, showing any two matchings can be connected through a sequence of compatible matchings.

Findings

The compatible bichromatic matching graph is connected.

Any two bichromatic matchings can be transformed into each other via compatible matchings.

The result answers an open problem in geometric graph theory.

Abstract

For a set $R$ of $n$ red points and a set $B$ of $n$ blue points, a $BR$-matching is a non-crossing geometric perfect matching where each segment has one endpoint in $B$ and one in $R$. Two $BR$-matchings are compatible if their union is also non-crossing. We prove that, for any two distinct $BR$-matchings $M$ and $M'$, there exists a sequence of $BR$-matchings $M = M_1, ..., M_k = M'$ such that $M_{i-1} $ is compatible with $M_i$. This implies the connectivity of the compatible bichromatic matching graph containing one node for each bichromatic matching and an edge joining each pair of compatible matchings, thereby answering the open problem posed by Aichholzer et al. in "Compatible matchings for bichromatic plane straight-line graphs"