## Abstract

Let L be a sequence (ℓ_{1}, ℓ_{2}, … , ℓ_{n}) of n lines in C^{3}. We define the intersection graphG_{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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Pages (from-to) | 986-1009 |

Number of pages | 24 |

Journal | Discrete and Computational Geometry |

Volume | 58 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2017 |

Externally published | Yes |

### Bibliographical note

Funding Information:Work on this paper was supported by Grant 892/13 from the Israel Science Foundation, by the Israeli Centers of Research Excellence (I-CORE) program (Center No. 4/11), and by a Shulamit Aloni Fellowship from the Israeli Ministry of Science.

Publisher Copyright:

© 2017, Springer Science+Business Media, LLC.

## Keywords

- Elekes–Sharir framework
- Graph rigidity
- Line configurations