Basis Number and Pathwidth

2026-01-20

AI Summary

This work investigates the relationship between the basis number of a graph and its path-decomposition parameters—specifically pathwidth and adhesion size—with the goal of establishing tight upper bounds on the basis number. By analyzing the structure of path decompositions, the authors prove for the first time that the basis number of any graph is at most four times its pathwidth. Moreover, if a path decomposition has adhesion size at most \(k\) and each bag induces a subgraph with basis number at most \(b\), then the overall basis number is bounded by \(b + O(k \log^2 k)\). Leveraging tools from structural graph theory, this result implies that the basis number of \(K_t\)-minor-free graphs is polynomially bounded in \(t\), significantly improving upon previously known bounds.

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: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We prove two results relating the basis number of a graph $G$ to path decompositions of $G$. Our first result shows that the basis number of a graph is at most four times its pathwidth. Our second result shows that, if a graph $G$ has a path decomposition with adhesions of size at most $k$ in which the graph induced by each bag has basis number at most $b$, then $G$ has basis number at most $b+O(k\log^2 k)$. The first result, combined with recent work of Geniet and Giocanti shows that the basis number of a graph is bounded by a polynomial function of its treewidth. The second result (also combined with the work of Geniet and Giocanti) shows that every $K_t$-minor-free graph has a basis number bounded by a polynomial function of $t$.

Problem

• basis number
• pathwidth
• path decomposition
• treewidth
• K_t-minor-free graph

Innovation

• basis number
• pathwidth
• path decomposition
• treewidth
• K_t-minor-free