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Separators for Intersection Graphs of Spheres

Authors Jacob Fox , Jonathan Tidor

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LIPIcs.SoCG.2026.50.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph theory
Keywords
  • graph separators
  • intersection graphs

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Abstract

We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension d. One of our results is that if an intersection graph of n spheres in ℝ^d has m edges, then it contains a balanced separator of size O_d(m^{1/d}n^{1-2/d}). This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.

Cite As Get BibTex

Jacob Fox and Jonathan Tidor. Separators for Intersection Graphs of Spheres. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.50

Author Details

Jacob Fox
  • Department of Mathematics, Stanford University, Stanford, CA, USA
Jonathan Tidor
  • Department of Mathematics, Princeton University, NJ, USA

Funding

This material is based in part upon work supported by the National Science Foundation under Grant No. DMS-1928930, while the authors were in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the Spring 2025 semester.
  • Fox, Jacob: supported by NSF Awards DMS-2452737 and DMS-2154129.
  • Tidor, Jonathan: supported in part by a Stanford Science Fellowship.

References

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