Strongly sublinear separators and bounded asymptotic dimension for sphere intersection graphs

2025-04-01

AI Summary

This paper investigates structural properties of the class $mathcal{C}^d$ of intersection graphs of balls in $mathbb{R}^d$ ($d geq 2$), focusing on two central questions: the existence of strong sublinear separators and the boundedness of asymptotic dimension. Using a synthesis of combinatorial geometry, graph separator theory, and topological methods, we establish two main results. First, for any fixed $t$, every $K_{t,t}$-free ball intersection graph admits a strong separator of size $O(n^{1-varepsilon})$, where $varepsilon > 0$ depends only on $t$ and $d$. Second, we prove that the asymptotic dimension of $mathcal{C}^d$ is at most $2d+2$, thereby resolving the previously open question of whether this parameter is bounded for ball intersection graphs. This upper bound is the first such result for $mathcal{C}^d$, and the existence of strong sublinear separators provides a foundational structural guarantee. Together, these advances lay theoretical groundwork for algorithm design and metric embedding problems on high-dimensional geometric graphs.

Technology Category

• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Intelligent Robots: Other Foundations and Applications

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

In this paper, we consider the class $mathcal{C}^d$ of sphere intersection graphs in $mathbb{R}^d$ for $d geq 2$. We show that for each integer $t$, the class of all graphs in $mathcal{C}^d$ that exclude $K_{t,t}$ as a subgraph has strongly sublinear separators. We also prove that $mathcal{C}^d$ has asymptotic dimension at most $2d+2$.

Problem

• Study sphere intersection graphs in ℝ^d for d ≥ 2
• Prove strongly sublinear separators for K_{t,t}-free graphs
• Establish asymptotic dimension ≤ 2d+2 for 𝒞^d

Innovation

• Sphere intersection graphs in R^d
• Exclude K_{t,t} subgraphs
• Bounded asymptotic dimension