Approximating branchwidth on parametric extensions of planarity

TL;DR

This paper extends polynomial-time approximation algorithms for branchwidth from planar graphs to certain minor-closed classes, specifically graphs excluding certain torus- and projective-plane-embeddable minors, using a novel decomposition approach.

Contribution

It introduces a method to approximate branchwidth in minor-closed graph classes beyond planar graphs by constructing a planar-torso decomposition with bounded difference in branchwidth.

Findings

Provides a constant-additive approximation algorithm for branchwidth.

Shows that minor-closed classes excluding certain embeddings admit efficient decompositions.

Extends the applicability of the Ratcatcher algorithm to broader graph classes.

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 $b+c$.