🤖 AI Summary
This paper addresses the branchwidth approximation problem for graph classes excluding two fixed graphs $H_1$ and $H_2$, each embeddable on the torus or the projective plane. For this broad family of non-planar graphs, we extend the Seymour–Thomas Ratcatcher algorithm—previously applicable only to planar graphs—to handle toroidal and projective-planar forbidden minors. Our method integrates the Graph Minor Structure Theorem, extraction of planar subgraphs, and constructive tree decomposition. The resulting algorithm runs in $O(|V|^3)$ time and achieves a constant additive approximation guarantee: the error depends solely on $H_1$ and $H_2$, not on the input graph size. This work overcomes a fundamental bottleneck—the intractability of exact branchwidth computation beyond planar graphs—and provides the first polynomial-time constant-additive approximation algorithm for branchwidth with rigorous theoretical guarantees on a wide class of non-planar graphs.
📝 Abstract
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 $b+c$.