### Abstract

In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x,y in P are adjacent whenever the open rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.

Original language | English (US) |
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Title of host publication | Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings |

Pages | 73-85 |

Number of pages | 13 |

Volume | 6844 LNCS |

DOIs | |

State | Published - 2011 |

Event | 12th International Symposium on Algorithms and Data Structures, WADS 2011 - New York, NY, United States Duration: Aug 15 2011 → Aug 17 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 6844 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 12th International Symposium on Algorithms and Data Structures, WADS 2011 |
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Country | United States |

City | New York, NY |

Period | 8/15/11 → 8/17/11 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings*(Vol. 6844 LNCS, pp. 73-85). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6844 LNCS). https://doi.org/10.1007/978-3-642-22300-6_7

**Witness rectangle graphs.** / Aronov, Boris; Dulieu, Muriel; Hurtado, Ferran.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings.*vol. 6844 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6844 LNCS, pp. 73-85, 12th International Symposium on Algorithms and Data Structures, WADS 2011, New York, NY, United States, 8/15/11. https://doi.org/10.1007/978-3-642-22300-6_7

}

TY - GEN

T1 - Witness rectangle graphs

AU - Aronov, Boris

AU - Dulieu, Muriel

AU - Hurtado, Ferran

PY - 2011

Y1 - 2011

N2 - In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x,y in P are adjacent whenever the open rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.

AB - In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x,y in P are adjacent whenever the open rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.

UR - http://www.scopus.com/inward/record.url?scp=80052128625&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052128625&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-22300-6_7

DO - 10.1007/978-3-642-22300-6_7

M3 - Conference contribution

SN - 9783642222993

VL - 6844 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 73

EP - 85

BT - Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings

ER -