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Structure and Independence in Hyperbolic Uniform Disk Graphs

Authors Thomas Bläsius , Jean-Pierre von der Heydt , Sándor Kisfaludi-Bak , Marcus Wilhelm , Geert van Wordragen

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ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational geometry
  • Theory of computation → Graph algorithms analysis
Keywords
  • hyperbolic geometry
  • unit disk graphs
  • independent set
  • treewidth

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Abstract

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.

Cite As Get BibTex

Thomas Bläsius, Jean-Pierre von der Heydt, Sándor Kisfaludi-Bak, Marcus Wilhelm, and Geert van Wordragen. Structure and Independence in Hyperbolic Uniform Disk Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.21

Author Details

Thomas Bläsius
  • Karlsruhe Institute of Technology, Germany
Jean-Pierre von der Heydt
  • Karlsruhe Institute of Technology, Germany
Sándor Kisfaludi-Bak
  • Aalto University, Espoo, Finland
Marcus Wilhelm
  • Karlsruhe Institute of Technology, Germany
Geert van Wordragen
  • Aalto University, Espoo, Finland

Funding

  • von der Heydt, Jean-Pierre: Funded by the pilot program Core-Informatics of the Helmholtz Association (HGF).
  • Kisfaludi-Bak, Sándor: Supported by the Research Council of Finland, Grant 363444.
  • Wilhelm, Marcus: funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 524989715.

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