The branchwidth of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher'' algorithm of Seymour and Thomas. We explore how this algorithm can be extended to minor-closed graph classes beyond planar graphs, as follows: Let $H_{1}$ be a graph embeddable in the torus and $H_{2}$ be a graph embeddable in the projective plane. We prove that every $\{H_{1},H_{2}\}$-minor free graph $G$ contains a subgraph $G'$ whose branchwidth differs from that of $G$ by a constant depending only on $H_1$ and $H_2$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition allows for a constant-additive approximation of branchwidth: For $\{H_{1},H_{2}\}$-minor free graphs, there is a constant $c$ (depending on $H_{1}$ and $H_{2}$) and an $\mathcal{O}(|V(G)|^{3})$-time algorithm that, given a graph $G$, outputs a value $b$ such that the branchwidth of $G$ is between $b$ and $b+c$.

翻译：分支宽度由 Robertson 和 Seymour 引入，作为图树可分解性的一种度量，与树宽互为替代。对于平面图，分支宽度可通过 Seymour 和 Thomas 著名的“捕鼠器”算法在多项式时间内计算。本文探讨如何将该算法推广到平面图之外的闭子式图类，具体如下：设 $H_{1}$ 为可嵌入环面的图，$H_{2}$ 为可嵌入射影平面的图。我们证明每个 $\{H_{1},H_{2}\}$-子式自由图 $G$ 均包含一个子图 $G'$，其分支宽度与 $G$ 的分支宽度相差仅依赖于 $H_1$ 和 $H_2$ 的常数。此外，图 $G'$ 允许一种树分解，其中所有 torso 均为平面图。该分解使得分支宽度的常数加性近似成为可能：对于 $\{H_{1},H_{2}\}$-子式自由图，存在常数 $c$（依赖于 $H_{1}$ 和 $H_{2}$）及一个 $\mathcal{O}(|V(G)|^{3})$ 时间算法，该算法在输入图 $G$ 后输出值 $b$，使得 $G$ 的分支宽度介于 $b$ 与 $b+c$ 之间。