## Abstract

Let S ⊆ R 2 be a set of n planar sites, such that each s ∈ S has an associated radius rs < 0. Let D(S) be the disk intersection graph for S. It has vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii rs, rt intersect. Our goal is to design data structures that maintain the connectivity structure of D(S) as sites are inserted and/or deleted. First, we consider unit disk graphs, i.e., rs = 1, for all s ∈ S. We describe a data structure that has O(log2 n) amortized update and O(log n/ log log n) amortized query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ rs ≤ Ψ, for a Ψ ≥ 1 known in advance. In the fully dynamic case, we achieve amortized update time O(Ψλ6(log n)log7 n) and query time O(log n/ log log n), where λs(n) is the maximum length of a Davenport-Schinzel sequence of order s on n symbols. In the incremental case, where only insertions are allowed, we get logarithmic dependency on Ψ, with O(α(n)) query time and O(log Ψλ6(log n)log7 n) update time. For the decremental setting, where only deletions are allowed, we first develop an efficient disk revealing structure: given two sets R and B of disks, we can delete disks from R, and upon each deletion, we receive a list of all disks in B that no longer intersect the union of R. Using this, we get decremental data structures with amortized query time O(log n/ log log n) that support m deletions in O((n log5 n + m log7 n)λ6(log n) + n log Ψ log4 n) overall time for bounded radius ratio Ψ and O((n log6 n + m log8 n)λ6(log n)) for arbitrary radii.

Original language | English |
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Title of host publication | 38th International Symposium on Computational Geometry, SoCG 2022 |

Editors | Xavier Goaoc, Michael Kerber |

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

ISBN (Electronic) | 9783959772273 |

DOIs | |

State | Published - 1 Jun 2022 |

Event | 38th International Symposium on Computational Geometry, SoCG 2022 - Berlin, Germany Duration: 7 Jun 2022 → 10 Jun 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 224 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 38th International Symposium on Computational Geometry, SoCG 2022 |
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Country/Territory | Germany |

City | Berlin |

Period | 7/06/22 → 10/06/22 |

### Bibliographical note

Publisher Copyright:© Haim Kaplan, Alexander Kauer, Katharina Klost, Kristin Knorr, Wolfgang Mulzer, Liam Roditty, and Paul Seiferth; licensed under Creative Commons License CC-BY 4.0

### Funding

Funding Supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF). Haim Kaplan: Partially supported by ISF grant 1595/19 and by the Blavatnik research foundation. Alexander Kauer: Supported in part by grant 1367/2016 from the German-Israeli Science Foundation (GIF), by the German Research Foundation within the collaborative DACH project Arrangements and Drawings as DFG Project MU 3501/3-1, and by ERC StG 757609. Kristin Knorr: Supported by the German Science Foundation within the research training group “Facets of Complexity” (GRK 2434). Wolfgang Mulzer: Supported in part by ERC StG 757609.

Funders | Funder number |
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Blavatnik research foundation | |

German–Israeli Science Foundation | |

Horizon 2020 Framework Programme | 757609 |

European Commission | |

Deutsche Forschungsgemeinschaft | MU 3501/3-1, GRK 2434 |

Israel Science Foundation | 1595/19 |

## Keywords

- Connectivity
- Disk Graphs
- Lower Envelopes