Separators for intersection graphs of spheres

2026-03-23

AI Summary

This work investigates the construction of balanced separators in intersection graphs of balls (or spheres) in high-dimensional Euclidean space. By integrating tools from combinatorial geometry and graph partitioning theory, it establishes—for the first time—the existence of a balanced separator of size \(O_d(m^{1/d} n^{1-2/d})\) in any intersection graph with \(n\) balls in \(\mathbb{R}^d\) and \(m\) edges, for arbitrary dimension \(d\). This bound is tight up to constant factors depending only on \(d\), resolving a long-standing open question regarding separator sizes in high-dimensional ball intersection graphs. Moreover, the result extends to fat convex objects and their boundary variants, substantially broadening the scope of existing separator theorems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

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 $\mathbb{R}^d$ 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.

Problem

• intersection graphs
• spheres
• separators
• fat convex bodies
• balanced separator

Innovation

• intersection graphs
• sphere separators
• balanced separator
• fat convex bodies
• geometric graph theory