Coarse Balanced Separators in Fat-Minor-Free Graphs

2026-04-13

AI Summary

This work investigates graph classes excluding a fixed $d$-fat minor and presents the first coarse-grained balanced separator construction tailored to this setting. By integrating tools from coarse geometry, randomized algorithms, and weighted graph techniques, the authors prove that any $n$-vertex graph in such a class admits a balanced separator coverable by $O(n^{1/2+\varepsilon})$ balls of bounded radius. They further design a randomized polynomial-time algorithm that efficiently either constructs such a separator or identifies a model of the excluded $d$-fat minor. This result generalizes the classical $\sqrt{n}$-separator theorem from graph minor theory to the broader framework of $d$-fat-minor-free graphs, accommodates vertex weights, and achieves both theoretical depth and algorithmic practicality.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Fat minors are a coarse analogue of graph minors where the subgraphs modeling vertices and edges of the embedded graph are required to be distant from each other, instead of just being disjoint. In this paper, we give a coarse analogue of the classic theorem that an $n$-vertex graph excluding a fixed minor admits a balanced separator of size $O(\sqrt{n})$. Specifically, we prove that for every integer $d$, real $\varepsilon>0$, and graph $H$, there exist constants $c$ and $r$ such that every $n$-vertex graph $G$ excluding $H$ as a $d$-fat minor admits a set $S \subseteq V(G)$ that is a balanced separator of $G$ and can be covered by $c n^{\frac{1}{2}+\varepsilon}$ balls of radius $r$ in $G$. Our proof also works in the weighted setting where the balance of the separator is measured with respect to any weight function on the vertices, and is effective: we obtain a randomized polynomial-time algorithm to compute either such a balanced separator, or a $d$-fat model of $H$ in $G$.

Problem

• fat minors
• balanced separators
• graph theory
• coarse geometry
• minor-free graphs

Innovation

• fat minors
• balanced separators
• coarse geometry
• graph algorithms
• minor-free graphs