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 $t$.

翻译：我们证明了关于图$G$的基数与$G$的路径分解之间的两个结果。我们的第一个结果表明，图的基数至多为其路径宽度的四倍。第二个结果表明，如果图$G$存在一个路径分解，其中粘连集大小至多为$k$，且每个包所诱导的子图的基数至多为$b$，则$G$的基数至多为$b+O(k\log^2 k)$。结合Geniet和Giocanti的最新工作，第一个结果证明了图的基数由其树宽的多项式函数界定。第二个结果（同样结合Geniet和Giocanti的工作）表明，每个不含$K_t$子式的图的基数可由$t$的多项式函数界定。