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.

@InProceedings{deberg_et_al:LIPIcs.ISAAC.2021.22,
  author =	{de Berg, Mark and Kisfaludi-Bak, S\'{a}ndor and Monemizadeh, Morteza and Theocharous, Leonidas},
  title =	{{Clique-Based Separators for Geometric Intersection Graphs}},
  booktitle =	{32nd International Symposium on Algorithms and Computation (ISAAC 2021)},
  pages =	{22:1--22:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-214-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{212},
  editor =	{Ahn, Hee-Kap and Sadakane, Kunihiko},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/15455},
  URN =		{urn:nbn:de:0030-drops-154556},
  doi =		{10.4230/LIPIcs.ISAAC.2021.22},
  annote =	{Keywords: Computational geometry, intersection graphs, separator theorems}
}