Basis Number of Graphs Excluding Minors

TL;DR

This paper proves that graphs excluding a fixed minor have bounded basis number, extending classical results and using advanced graph decomposition techniques, with implications for understanding cycle structures in complex graphs.

Contribution

It establishes that graphs excluding a fixed minor have bounded basis number, generalizing Mac Lane's theorem and employing the Graph Minor Structure Theorem and tree-decomposition analysis.

Findings

Graphs of treewidth k have basis number bounded by a function of k.

Graphs excluding a fixed minor have basis number bounded by a polynomial in |H|.

The proof framework adapts techniques from Courcelle's conjecture and related results.

Abstract

The basis number of a graph $G$ is the minimum $k$ such that the cycle space of $G$ is generated by a family of cycles using each edge at most $k$ times. A classical result of Mac Lane states that planar graphs are exactly graphs with basis number at most 2, and more generally, graphs embedded on a fixed surface of bounded genus are known to have bounded basis number. Generalising this, we prove that graphs excluding a fixed minor $H$ have bounded basis number. Our proof uses the Graph Minor Structure Theorem, which requires us to understand how basis number behaves in tree-decompositions. In particular, we prove that graphs of treewidth $k$ have basis number bounded by some function of $k$. We handle tree-decompositions using the proof framework developed by Boja\'nczyk and Pilipczuk in their proof of Courcelle's conjecture. Combining our approach with independent results of Miraftab, Morin and Yuditsky (2025) on basis number and path-decompositions, one can moreover improve our upper bound to a polynomial one: there exists an absolute constant $c>0$ such that every $H$-minor free graph has basis number $O(|H|^c)$.