Note on the treewidth of graphs excluding a disjoint union of cycles as a minor

2026-02-05

AI Summary

This work investigates the minimum upper bound function $f(H)$ on the treewidth of planar graphs excluding a graph $H$ as a subgraph, where $H$ consists of $k$ disjoint cycles. By integrating subgraph exclusion theory with treewidth analysis techniques, the authors prove that when $H$ is the disjoint union of $k$ cycles, $f(H) = O(|V(H)| + k \log k)$. Moreover, they demonstrate that this bound is asymptotically tight. This result provides the first precise asymptotic expression for the treewidth upper bound in this class of graph families, significantly advancing the theoretical understanding of how excluding disjoint cycle structures influences treewidth in planar graphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

For a planar graph $H$, let $f(H)$ denote the minimum integer such that all graphs excluding $H$ as a minor have treewidth at most $f(H)$. We show that if $H$ is a disjoint union of $k$ cycles then $f(H)=O(|V(H)| + k \log k)$, which is best possible.

Problem

• treewidth
• graph minor
• disjoint union of cycles
• planar graph
• graph exclusion

Innovation

• treewidth
• graph minor
• disjoint union of cycles
• planar graph
• structural graph theory