eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-11-30
22:1
22:15
10.4230/LIPIcs.ISAAC.2021.22
article
Clique-Based Separators for Geometric Intersection Graphs
de Berg, Mark
1
Kisfaludi-Bak, Sándor
2
Monemizadeh, Morteza
1
Theocharous, Leonidas
1
Department of Computer Science, TU Eindhoven, The Netherlands
Institute for Theoretical Studies, ETH Zürich, Switzerland
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol212-isaac2021/LIPIcs.ISAAC.2021.22/LIPIcs.ISAAC.2021.22.pdf
Computational geometry
intersection graphs
separator theorems