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 investigate an extension of this algorithm to minor-closed graph classes, further than 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 $G'$ where the difference between the branchwidth of $G$ and the branchwidth of $G'$ is bounded by some constant, depending only on $H_{1}$ and $H_{2}$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving a constant-additive approximation for branchwidth: For $\{H_{1},H_{2}\}$-minor free graphs, there is a constant $c$ (depending on $H_{1}$ and $H_{2}$) and an $\Ocal(|V(G)|^{3})$-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：图的**分支宽度**由Robertson和Seymour引入，作为衡量图树可分解性的指标，是树宽的替代度量。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。本文研究将该算法推广到比平面图更广的、闭于子式操作的图类，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$G$均包含子图$G'$，使得$G$的分支宽度与$G'$的分支宽度之差受限于仅依赖于$H_{1}$和$H_{2}$的常数。此外，$G'$允许一种树分解，其中所有torso均为平面图。该分解可用于推导分支宽度的常数加性近似：对于$\{H_{1},H_{2}\}$-子式自由图，存在常数$c$（依赖于$H_{1}$和$H_{2}$）以及$\Ocal(|V(G)|^{3})$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。