## Abstract

Let S ⊂ ℝ^{2} be a set of n sites, where each s ∈ S has an associated radius r_{s} > 0. The disk graph D(S) is the undirected graph with vertex set S and an undirected edge between two sites s, t ∈ S if and only if |st| ≤ r_{s} + r_{t}, i.e., if the disks with centers s and t and respective radii rs and rt intersect. Disk graphs are used to model sensor networks. Similarly, the transmission graph T(S) is the directed graph with vertex set S and a directed edge from a site s to a site t if and only if |st| ≤ rs, i.e., if t lies in the disk with center s and radius rs. We provide algorithms for detecting (directed) triangles and, more generally, computing the length of a shortest cycle (the girth) in D(S) and in T(S). These problems are notoriously hard in general, but better solutions exist for special graph classes such as planar graphs. We obtain similarly efficient results for disk graphs and for transmission graphs. More precisely, we show that a shortest (Euclidean) triangle in D(S) and in T(S) can be found in O(n log n) expected time, and that the (weighted) girth of D(S) can be found in O(n log n) expected time. For this, we develop new tools for batched range searching that may be of independent interest.

Original language | English |
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Title of host publication | 27th Annual European Symposium on Algorithms, ESA 2019 |

Editors | Michael A. Bender, Ola Svensson, Grzegorz Herman |

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

ISBN (Electronic) | 9783959771245 |

DOIs | |

State | Published - Sep 2019 |

Event | 27th Annual European Symposium on Algorithms, ESA 2019 - Munich/Garching, Germany Duration: 9 Sep 2019 → 11 Sep 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 144 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 27th Annual European Symposium on Algorithms, ESA 2019 |
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Country/Territory | Germany |

City | Munich/Garching |

Period | 9/09/19 → 11/09/19 |

### Bibliographical note

Publisher Copyright:© Haim Kaplan, Katharina Klost, Wolfgang Mulzer, Liam Roditty, Paul Seiferth, and Micha Sharir.

## Keywords

- Disk graph
- Girth
- Transmission graph
- Triangle