2026-07-22T15:57:09Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23173 2025-10-02T12:41:54Z ddc:004 open_access

Structure and Independence in Hyperbolic Uniform Disk Graphs Bläsius, Thomas von der Heydt, Jean-Pierre Kisfaludi-Bak, Sándor Wilhelm, Marcus van Wordragen, Geert hyperbolic geometry unit disk graphs independent set treewidth We consider intersection graphs of disks of radius r in the hyperbolic plane. Unlike the Euclidean setting, these graph classes are different for different values of r, where very small r corresponds to an almost-Euclidean setting and r ∈ Ω(log n) corresponds to a firmly hyperbolic setting. We observe that larger values of r create simpler graph classes, at least in terms of separators and the computational complexity of the Independent Set problem. First, we show that intersection graphs of disks of radius r in the hyperbolic plane can be separated with 𝒪((1+1/r)log n) cliques in a balanced manner. Our second structural insight concerns Delaunay complexes in the hyperbolic plane and may be of independent interest. We show that for any set S of n points with pairwise distance at least 2r in the hyperbolic plane, the corresponding Delaunay complex has outerplanarity 1+𝒪((log n)/r), which implies a similar bound on the balanced separators and treewidth of such Delaunay complexes. Using this outerplanarity (and treewidth) bound we prove that Independent Set can be solved in n^𝒪(1+(log n)/r) time. The algorithm is based on dynamic programming on some unknown sphere cut decomposition that is based on the solution. The resulting algorithm is a far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is tight under the Exponential Time Hypothesis. In particular, Independent Set is polynomial-time solvable in the firmly hyperbolic setting of r ∈ Ω(log n). Finally, in the case when the disks have ply (depth) at most 𝓁, we give a PTAS for Maximum Independent Set that has only quasi-polynomial dependence on 1/ε and 𝓁. Our PTAS is a further generalization of our exact algorithm. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Thomas Bläsius and Jean-Pierre von der Heydt and Sándor Kisfaludi-Bak and Marcus Wilhelm and Geert van Wordragen 2025 Is Part Of LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2025.21 urn:nbn:de:0030-drops-231731 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.21 eng https://creativecommons.org/licenses/by/4.0/legalcode