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 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$G$的基环数是指使得$G$的环空间可由一族环生成的最小整数$k$，其中每条边至多被使用$k$次。Mac Lane的经典结论表明，平面图恰好是基环数不超过2的图；更一般地，已知嵌入在固定亏格曲面上的图具有有界基环数。作为推广，我们证明了排除固定子式$H$的图具有有界基环数。我们的证明使用了图子式结构定理，这要求我们理解基环数在树分解中的行为。特别地，我们证明了树宽为$k$的图的基环数可由$k$的某个函数界定。我们利用Bojańczyk与Pilipczuk在证明Courcelle猜想时建立的证明框架来处理树分解。将我们的方法与Miraftab、Morin及Yuditsky（2025年）关于基环数与路径分解的独立结果相结合，可进一步将上界改进为多项式形式：存在绝对常数$c>0$，使得每个排除$H$子式的图具有基环数$O(|H|^c)$。