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

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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.

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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

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.

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