### Abstract

This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are always at least n-22 edges, where n=|A|+|B|, which is tight in the worst case. We also show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable.

Original language | English (US) |
---|---|

Pages (from-to) | 519-523 |

Number of pages | 5 |

Journal | Information Processing Letters |

Volume | 114 |

Issue number | 10 |

DOIs | |

State | Published - 2014 |

### Fingerprint

### Keywords

- Computational geometry
- Delaunay graphs
- Gabriel graphs
- Proximity graphs
- Rectangle of influence graphs
- Witness graphs

### ASJC Scopus subject areas

- Information Systems
- Computer Science Applications
- Signal Processing
- Theoretical Computer Science

### Cite this

*Information Processing Letters*,

*114*(10), 519-523. https://doi.org/10.1016/j.ipl.2014.04.001

**Mutual witness proximity graphs.** / Aronov, Boris; Dulieu, Muriel; Hurtado, Ferran.

Research output: Contribution to journal › Article

*Information Processing Letters*, vol. 114, no. 10, pp. 519-523. https://doi.org/10.1016/j.ipl.2014.04.001

}

TY - JOUR

T1 - Mutual witness proximity graphs

AU - Aronov, Boris

AU - Dulieu, Muriel

AU - Hurtado, Ferran

PY - 2014

Y1 - 2014

N2 - This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are always at least n-22 edges, where n=|A|+|B|, which is tight in the worst case. We also show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable.

AB - This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are always at least n-22 edges, where n=|A|+|B|, which is tight in the worst case. We also show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable.

KW - Computational geometry

KW - Delaunay graphs

KW - Gabriel graphs

KW - Proximity graphs

KW - Rectangle of influence graphs

KW - Witness graphs

UR - http://www.scopus.com/inward/record.url?scp=84899854890&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84899854890&partnerID=8YFLogxK

U2 - 10.1016/j.ipl.2014.04.001

DO - 10.1016/j.ipl.2014.04.001

M3 - Article

AN - SCOPUS:84899854890

VL - 114

SP - 519

EP - 523

JO - Information Processing Letters

JF - Information Processing Letters

SN - 0020-0190

IS - 10

ER -