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A Clique-Based Separator for Intersection Graphs of Geodesic Disks in ℝ²

Authors Boris Aronov , Mark de Berg , Leonidas Theocharous

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Author Details

Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Leonidas Theocharous
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands

Funding

  • Aronov, Boris: Work has been supported by NSF grant CCF 20-08551.
  • de Berg, Mark: MdB is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.
  • Theocharous, Leonidas: LT is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.

Cite AsGet BibTex

Boris Aronov, Mark de Berg, and Leonidas Theocharous. A Clique-Based Separator for Intersection Graphs of Geodesic Disks in ℝ². In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.9

Abstract

Let d be a (well-behaved) shortest-path metric defined on a path-connected subset of ℝ² and let 𝒟 = {D_1,…,D_n} be a set of geodesic disks with respect to the metric d. We prove that 𝒢^×(𝒟), the intersection graph of the disks in 𝒟, has a clique-based separator consisting of O(n^{3/4+ε}) cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for q-Coloring that runs in time 2^O(n^{3/4+ε}), assuming the boundaries of the disks D_i can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses O(n^{7/4+ε}) storage and can report the hop distance between any two nodes in 𝒢^×(𝒟) in O(n^{3/4+ε}) time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational geometry
  • intersection graphs
  • separator theorems

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