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{枝宽}由Roberson和Seymour引入，作为衡量图树分解性的另一种指标，其替代树宽。枝宽在平面图上可通过Seymour与Thomas著名的"Ratcatcher"算法在多项式时间内计算。本文将该算法推广至比平面图更广泛的闭子图类：设$H_{0}$为可嵌入射影平面的图，$H_{1}$为可嵌入环面的图。我们证明每个不含$\{H_{0},H_{1}\}$作为子式的图$G$均存在子图$G'$，使得$G$与$G'$的枝宽之差由仅依赖于$H_{0}$和$H_{1}$的常数界定。此外，$G'$具有所有躯干均为平面图的树分解。该分解可用于推导枝宽的EPTAS：对于不含$\{H_{0},H_{1}\}$作为子式的图，存在函数$f\colon\mathbb{N}\to\mathbb{N}$及枝宽的$(1+\epsilon)$-近似算法，对任意$\epsilon>0$，运行时间为$\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n)$。