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 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 Bojańczyk 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 $O(|H|^c)$.

翻译：图$G$的基数是指使得$G$的圈空间可由一族圈生成的最小整数$k$，其中每条边至多被使用$k$次。Mac Lane的一个经典结论指出，平面图恰好是基数至多为2的图；更一般地，已知嵌入在固定曲面上的图具有有界基数。作为推广，我们证明了排除一个固定子图$H$的图具有有界基数。我们的证明使用了图子式结构定理，这要求我们理解基数在树分解中的行为。特别地，我们证明了树宽为$k$的图的基数由$k$的某个函数界定。我们利用Bojańczyk和Pilipczuk在证明Courcelle猜想时发展的证明框架来处理树分解。将我们的方法与Miraftab、Morin和Yuditsky（2025年）关于基数与路径分解的独立结果相结合，可以进一步将我们的上界改进为多项式界：存在一个绝对常数$c>0$，使得每个排除$H$子式的图的基数为$O(|H|^c)$。