Approximating branchwidth on parametric extensions of planarity

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 $b+c$.