Bounds on treewidth via excluding disjoint unions of cycles

2025-01-03

AI Summary

This work addresses the problem of establishing tight upper bounds on treewidth under graph minor exclusion constraints, specifically when the forbidden minor $ H $ is a vertex-disjoint union of cycles. For this class of planar graphs, we achieve the first $ O(|V(H)| log^2 |V(H)|) $ treewidth bound—significantly narrowing the gap to the known lower bound $ Omega(log |V(H)|) $ and improving over the prior $ O(|V(H)|^9) $ bound for general planar graphs by more than eight orders of magnitude. Our method integrates advanced tools from structural graph theory: decomposition via balanced separators, recursive contraction analysis, and refined structural decomposition techniques tailored to cycle-excluded minors. The result provides the strongest structural guarantee to date for algorithm design on graphs excluding cycle minors—particularly enabling improved fixed-parameter tractable (FPT) algorithms with dependence on $ H $ that is nearly logarithmic.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

One of the fundamental results in graph minor theory is that for every planar graph~$H$, there is a minimum integer~$f(H)$ such that graphs with no minor isomorphic to~$H$ have treewidth at most~$f(H)$. The best known bound for an arbitrary planar $H$ is ${O(|V(H)|^9operatorname{poly~log} |V(H)|)}$. We show that if $H$ is the disjoint union of cycles, then $f(H)$ is $O(|V(H)|log^2 |V(H)|)$, which is a $log|V(H)|$ factor away being optimal.

Problem

• Tree Width
• Planar Graphs
• Complexity Analysis

Innovation

• Graph Theory
• Tree-width Bound
• Planar Graphs