Clique-Based Separators for Geometric Intersection Graphs

Mark de Berg, Sándor Kisfaludi-Bak, Morteza Monemizadeh, Leonidas Theocharous (Corresponding author)

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3 Citations (Scopus)
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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 δn, for some fixed constant δ< 1. The weight of a clique-based separator is defined as ∑ CSlog (| C| + 1). Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 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. (i) Map graphs admit a balanced clique-based separator of weight O(n), which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n2 / 3log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(nlogn). (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n2 / 3log n). (iv) Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(n+rlog(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.

Original languageEnglish
Pages (from-to)1652-1678
Number of pages27
JournalAlgorithmica
Volume85
Issue number6
Early online date15 Oct 2022
DOIs
Publication statusPublished - Jun 2023

Bibliographical note

Publisher Copyright:
© 2022, The Author(s).

Keywords

  • Computational geometry
  • Intersection graphs
  • Separator theorems

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