## Abstract

Let L be a sequence (ℓ_{1}, ℓ_{2}, ..., ℓ_{n}) of n lines in double-struck C^{3}. We define the intersection graph G_{L} = ([n], E) of L, where [n] := {1, ..., n}, and with {i, j} ∈ E if and only if i ≠ j and the corresponding lines ℓ_{i} and ℓ_{j} intersect, or are parallel (or coincide). For a graph G = ([n], E), we say that a sequence L is a realization of G if G ⊂ G_{L}. One of the main results of this paper is to provide a combinatorial characterization of graphs G = ([n], E) that have the following property: For every generic (see Definition 4.1) realization L of G, that consists of n pairwise distinct lines, we have G_{L} = K_{n}, in which case the lines of L are either all concurrent or all coplanar. The general statements that we obtain about lines, apart from their independent interest, turns out to be closely related to the notion of graph rigidity. The connection is established due to the so-called Elekes-Sharir framework, which allows us to transform the problem into an incidence problem involving lines in three dimensions. By exploiting the geometry of contacts between lines in 3D, we can obtain alternative, simpler, and more precise characterizations of the rigidity of graphs.

Original language | American English |
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Title of host publication | 32nd International Symposium on Computational Geometry, SoCG 2016 |

Editors | Sandor Fekete, Anna Lubiw |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 58.1-58.14 |

ISBN (Electronic) | 9783959770095 |

DOIs | |

State | Published - 1 Jun 2016 |

Externally published | Yes |

Event | 32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States Duration: 14 Jun 2016 → 17 Jun 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 51 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 32nd International Symposium on Computational Geometry, SoCG 2016 |
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Country/Territory | United States |

City | Boston |

Period | 14/06/16 → 17/06/16 |

### Bibliographical note

Publisher Copyright:© Orit Esther Raz.

## Keywords

- Global rigidity
- Laman graphs
- Line configurations
- Rigidity