### Abstract

Let P=B∪R be a set of 2n points in general position in the plane, where B is a set of n blue points and R a set of n red points. A 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 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. At 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. Moreover, we present an O(n^{2}logn) time algorithm for constructing a transformation of length O(n) between two given BR-matchings.

Original language | English |
---|---|

Pages (from-to) | 77-88 |

Number of pages | 12 |

Journal | Computational Geometry: Theory and Applications |

Volume | 68 |

DOIs | |

Publication status | Published - 1 Mar 2018 |

### Fingerprint

### Keywords

- Bichromatic point set
- Compatible matchings
- Perfect matchings
- Reconfiguration problem
- Transformation graph

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computational Geometry: Theory and Applications*,

*68*, 77-88. https://doi.org/10.1016/j.comgeo.2017.05.003

**Linear transformation distance for bichromatic matchings.** / Aichholzer, Oswin; Barba, Luis; Hackl, Thomas; Pilz, Alexander; Vogtenhuber, Birgit.

Research output: Contribution to journal › Article › Research › peer-review

*Computational Geometry: Theory and Applications*, vol. 68, pp. 77-88. https://doi.org/10.1016/j.comgeo.2017.05.003

}

TY - JOUR

T1 - Linear transformation distance for bichromatic matchings

AU - Aichholzer, Oswin

AU - Barba, Luis

AU - Hackl, Thomas

AU - Pilz, Alexander

AU - Vogtenhuber, Birgit

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Let P=B∪R be a set of 2n points in general position in the plane, where B is a set of n blue points and R a set of n red points. A 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 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. At 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. Moreover, we present an O(n2logn) time algorithm for constructing a transformation of length O(n) between two given BR-matchings.

AB - Let P=B∪R be a set of 2n points in general position in the plane, where B is a set of n blue points and R a set of n red points. A 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 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. At 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. Moreover, we present an O(n2logn) time algorithm for constructing a transformation of length O(n) between two given BR-matchings.

KW - Bichromatic point set

KW - Compatible matchings

KW - Perfect matchings

KW - Reconfiguration problem

KW - Transformation graph

UR - http://www.scopus.com/inward/record.url?scp=85019743182&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2017.05.003

DO - 10.1016/j.comgeo.2017.05.003

M3 - Article

VL - 68

SP - 77

EP - 88

JO - Computational geometry

JF - Computational geometry

SN - 0925-7721

ER -