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Approximating branchwidth on parametric extensions of planarity

Dimitrios M. Thilikos, Sebastian Wiederrecht

TL;DR

This work extends the Ratcatcher paradigm from planar graphs to minor-closed classes that exclude a torus-embeddable and a projective-plane-embeddable graph, proving a constant-additive approximation for branchwidth. The authors introduce a combinatorial framework built on walls (handle and crosscap) and a structural decomposition where torsos become planar after removing a small apex set, enabling the use of planar branchwidth techniques within a single decomposition. A key technical device is the interplay between tangles and slopes on spherical embeddings, which yields an additive bound onbw when passing to near-planar components, and supports a polynomial-time (in an exponential-in-k sense) algorithm that either finds walls or produces the desired decomposition. Consequently, for any ${H_1} obreak ext{ in } obreak ext{E}_1$ and ${H_2} obreak ext{ in } obreak ext{E}_2$, there is an ${ m O}igl(2^{2^{ ext{poly}(k)}}|V(G)|^3igr)$-time procedure that outputs a value $b$ with ${ m bw}(G)\nobreak ext{ in }[b,b+c]$, where $c$ depends only on $|V(H_1)|+|V(H_2)|$, and a more refined ${ ext{EPTAS}}$-type result follows. This advances efficient branchwidth computation beyond planarity to broad minor-closed graph classes while preserving planarity-like algorithmic advantages on the decomposed pieces.

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 $b+c$.

Approximating branchwidth on parametric extensions of planarity

TL;DR

This work extends the Ratcatcher paradigm from planar graphs to minor-closed classes that exclude a torus-embeddable and a projective-plane-embeddable graph, proving a constant-additive approximation for branchwidth. The authors introduce a combinatorial framework built on walls (handle and crosscap) and a structural decomposition where torsos become planar after removing a small apex set, enabling the use of planar branchwidth techniques within a single decomposition. A key technical device is the interplay between tangles and slopes on spherical embeddings, which yields an additive bound onbw when passing to near-planar components, and supports a polynomial-time (in an exponential-in-k sense) algorithm that either finds walls or produces the desired decomposition. Consequently, for any and , there is an -time procedure that outputs a value with , where depends only on , and a more refined -type result follows. This advances efficient branchwidth computation beyond planarity to broad minor-closed graph classes while preserving planarity-like algorithmic advantages on the decomposed pieces.

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 be a graph embeddable in the torus and be a graph embeddable in the projective plane. We prove that every -minor free graph contains a subgraph whose branchwidth differs from that of by a constant depending only on and . Moreover, the graph admits a tree decomposition where all torsos are planar. This decomposition allows for a constant-additive approximation of branchwidth: For -minor free graphs, there is a constant (depending on and ) and an -time algorithm that, given a graph , outputs a value such that the branchwidth of is between and .
Paper Structure (15 sections, 17 theorems, 5 equations, 1 figure)

This paper contains 15 sections, 17 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Algorithms for branchwidth
  3. Computing branchwidth of planar graphs.
  4. Approximation algorithms of branchwidth.
  5. Approximating branchwidth
  6. Main combinatorial result
  7. Decompositions and algorithms
  8. Decompositions and tangles
  9. Branchwidth.
  10. Tangles.
  11. Approximation algorithms
  12. Proof for \ref{['maintscomb']}
  13. Almost spherical embeddings
  14. From tangles to slopes
  15. Conclusion

Key Result

Theorem 1.1

Branchwidth admits a polynomial-time additive approximation algorithm for every graph class that excludes as a minor a graph $H_{1}\in \mathcal{E}_{1}$ and a graph $H_{2}\in \mathcal{E}_{2}$. In particular, there is a function $f_{arst_cpe}\colon\mathbb{N}\to\mathbb{N}$ and an algorithm that, given

Figures (1)

  • Figure 1: The elementary annulus wall $\mathscr{W}_{9}^{\texttt{A}}$, the elementary handle wall $\mathscr{W}_{9}^{\texttt{H}}$, and the elementary crosscap wall $\mathscr{W}_{9}^{\texttt{C}}$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1: Tree decompositions
  • Proposition 2.3: Ratcatcher algorithm SeymourT94callr
  • Proposition 2.4: BodlaenderT97const
  • Lemma 2.5
  • Definition 2.6: Tangle
  • ...and 23 more