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Clique-Based Separators for Geometric Intersection Graphs

Authors Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, Leonidas Theocharous

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Author Details

Mark de Berg
  • Department of Computer Science, TU Eindhoven, The Netherlands
Sándor Kisfaludi-Bak
  • Institute for Theoretical Studies, ETH Zürich, Switzerland
Morteza Monemizadeh
  • Department of Computer Science, TU Eindhoven, The Netherlands
Leonidas Theocharous
  • Department of Computer Science, TU Eindhoven, The Netherlands

Funding

The work in this paper is supported by the Dutch Research Council (NWO) through Gravitation-grant NETWORKS-024.002.003.

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Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, and Leonidas Theocharous. Clique-Based Separators for Geometric Intersection Graphs. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 22:1-22:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.22

Abstract

Let F be a set of n objects in the plane and let 𝒢^{×}(F) be its intersection graph. A balanced clique-based separator of 𝒢^{×}(F) is a set 𝒮 consisting of cliques whose removal partitions 𝒢^{×}(F) into components of size at most δ n, for some fixed constant δ < 1. The weight of a clique-based separator is defined as ∑_{C ∈ 𝒮}log (|C|+1). Recently De Berg et al. (SICOMP 2020) proved that if S consists of convex fat objects, then 𝒢^{×}(F) admits a balanced clique-based separator of weight O(√n). We extend this result in several directions, obtaining the following results.  
- Map graphs admit a balanced clique-based separator of weight O(√n), which is tight in the worst case. 
- Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3} log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(√n log n). 
- Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3} log n). 
- Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(√n + r log(n/r)), which is tight in the worst case.  These results immediately imply sub-exponential algorithms for MAXIMUM INDEPENDENT SET (and, hence, VERTEX COVER), for FEEDBACK VERTEX SET, and for q-Coloring for constant q in these 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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