Bichromatic compatible matchings

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′M′, there exists a sequence of BR  -matchings M=M1,…,Mk=M′M=M1,…,Mk=M′ such that Mi−1Mi−1 is compatible with MiMi. This implies the connectivity of the compatible bichromatic matching graph containing one node for each BR-matching and an edge joining each pair of compatible BR-matchings, thereby answering the open problem posed by Aichholzer et al. in [1].