Approximating branchwidth on parametric extensions of planarity
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.