Product Structure and Treewidth of Hyperbolic Uniform Disk Graphs

2026-03-19

AI Summary

This study investigates whether hyperbolic unit disk graphs (HUDGs) exhibit grid-like product structures and explores the relationship between their treewidth and clique number. Employing techniques from graph products, treewidth analysis, clique covers, and constructions rooted in hyperbolic geometry, the work establishes that HUDGs with bounded clique number cannot be expressed as strong products of a constant-treewidth graph and a path, thereby revealing their inherently non-grid global structure. Furthermore, the authors construct explicit families of HUDGs with bounded clique number but unbounded treewidth, achieving a treewidth-to-clique-number ratio of $\Omega(\log n / \log\log n)$. They also disprove a conjecture asserting that balanced separators in such graphs admit clique covers of size less than $\log n$, demonstrating that both local and layered tree-independence numbers are unbounded.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Topic discovery and tracking

Abstract

Hyperbolic uniform disk graphs (HUDGs) are intersection graphs of disks with some radius $r$ in the hyperbolic plane, where $r$ may be constant or depend on the number of vertices in a family of HUDGs. We show that HUDGs with constant clique number do not admit \emph{product structure}, i.e., that there is no constant $c$ such that every such graph is a subgraph of $H \boxtimes P$ for some graph $H$ of treewidth at most $c$. This justifies that HUDGs are described as not having a grid-like structure in the literature, and is in contrast to unit disk graphs in the Euclidean plane, whose grid-like structure is evident from the fact that they are subgraphs of the strong product of two paths and a clique of constant size [Dvořák et al., '21, MATRIX Annals]. By allowing $H$ to be any graph of constant treewidth instead of a path-like graph, we reject the possibility of a grid-like structure not merely by the maximum degree (which is unbounded for HUDGs) but due to their global structure. We complement this by showing that for every (sub-)constant $r$, HUDGs admit product structure, whereas the typical hyperbolic behavior is observed if $r$ grows with the number of vertices. Our proof involves a family of $n$-vertex HUDGs with radius $\log n$ that has bounded clique number but unbounded treewidth, and one for which the ratio of treewidth and clique number is $\log n / \log \log n$. Up to a $\log \log n$ factor, this negatively answers a question raised by Bläsius et al. [SoCG '25] asking whether balanced separators of HUDGs with radius $\log n$ can be covered by less than $\log n$ cliques. Our results also imply that the local and layered tree-independence number of HUDGs are both unbounded, answering an open question of Dallard et al. [arXiv '25].

Problem

• Hyperbolic Uniform Disk Graphs
• product structure
• treewidth
• clique number
• grid-like structure

Innovation

• product structure
• treewidth
• hyperbolic uniform disk graphs
• clique number
• balanced separators