The \textsl{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 $H_{0}$ be a graph embeddedable in the projective plane and $H_{1}$ be a graph embeddedable in the torus. We prove that every $\{H_{0},H_{1}\}$-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 $H_{0}$ and $H_{1}$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving an EPTAS for branchwidth: For $\{H_{0},H_{1}\}$-minor free graphs, there is a function $f\colon\mathbb{N}\to\mathbb{N}$ and a $(1+\epsilon)$-approximation algorithm for branchwidth, running in time $\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n),$ for every $\epsilon>0$.

翻译：图的\textsl{分支宽度}由罗伯逊和西摩引入，作为衡量图树分解性的指标，是树宽的一种替代。对于平面图，分支宽度可通过西摩和托马斯著名的“捕鼠器”算法在多项式时间内计算。我们将此算法扩展到比平面图更广的、对子图封闭的图类，具体如下：设$H_{0}$为可嵌入射影平面的图，$H_{1}$为可嵌入环面的图。我们证明，每个$\{H_{0},H_{1}\}$-无子式图$G$均包含一个子图$G'$，使得$G$与$G'$的分支宽度之差受限于某个仅依赖于$H_{0}$和$H_{1}$的常数。此外，$G'$允许一种树分解，其中所有躯干均为平面图。该分解可用于推导分支宽度的EPTAS：对于$\{H_{0},H_{1}\}$-无子式图，存在函数$f\colon\mathbb{N}\to\mathbb{N}$及一个$(1+\epsilon)$-近似算法，其运行时间为$\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n)$，适用于任意$\epsilon>0$。