eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-06-06
9:1
9:15
10.4230/LIPIcs.SoCG.2024.9
article
A Clique-Based Separator for Intersection Graphs of Geodesic Disks in ℝ²
Aronov, Boris
1
https://orcid.org/0000-0003-3110-4702
de Berg, Mark
2
https://orcid.org/0000-0001-5770-3784
Theocharous, Leonidas
2
https://orcid.org/0000-0002-1707-6787
Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol293-socg2024/LIPIcs.SoCG.2024.9/LIPIcs.SoCG.2024.9.pdf
Computational geometry
intersection graphs
separator theorems