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On a tree-based variant of bandwidth and forbidding simple topological minors

Hugo Jacob, William Lochet, Christophe Paul

TL;DR

The paper advances the understanding of tree-like bandwidth by characterising graphs that exclude a $k$-fan or a $k$-dipole as a topological minor, and by introducing treebandwidth, the minimum bandwidth achievable under tree-layouts. It provides structure theorems yielding FPT-linear decompositions with controlled adhesion and torso growth, enabling an FPT-linear-time approximation for treebandwidth and a thorough complexity taxonomy for exact computation, including XALP membership and hardness results. The authors leverage SPQR trees to treat easy cases (e.g., forbidding a gem) and develop a robust framework linking topological minor obstructions to tree-based decompositions, with algorithmic consequences for recognition and approximation. The work also clarifies relationships between treebandwidth and related notions such as tree-partition-width and proper-chordal completions, situating tbw within a broader landscape of tree-like width parameters and graph-search characterisations. Overall, the results offer both theoretical structure theorems and practical, scalable algorithms for approximating treebandwidth while highlighting intrinsic parameterised complexity barriers for exact computation.

Abstract

We obtain structure theorems for graphs excluding a fan (a path with a universal vertex) or a dipole ($K_{2,k}$) as a topological minor. The corresponding decompositions can be computed in FPT linear time. This is motivated by the study of a graph parameter we call treebandwidth which extends the graph parameter bandwidth by replacing the linear layout by a rooted tree such that neighbours in the graph are in ancestor-descendant relation in the tree. We deduce an approximation algorithm for treebandwidth running in FPT linear time from our structure theorems. We complement this result with a precise characterisation of the parameterised complexity of computing the parameter exactly.

On a tree-based variant of bandwidth and forbidding simple topological minors

TL;DR

The paper advances the understanding of tree-like bandwidth by characterising graphs that exclude a -fan or a -dipole as a topological minor, and by introducing treebandwidth, the minimum bandwidth achievable under tree-layouts. It provides structure theorems yielding FPT-linear decompositions with controlled adhesion and torso growth, enabling an FPT-linear-time approximation for treebandwidth and a thorough complexity taxonomy for exact computation, including XALP membership and hardness results. The authors leverage SPQR trees to treat easy cases (e.g., forbidding a gem) and develop a robust framework linking topological minor obstructions to tree-based decompositions, with algorithmic consequences for recognition and approximation. The work also clarifies relationships between treebandwidth and related notions such as tree-partition-width and proper-chordal completions, situating tbw within a broader landscape of tree-like width parameters and graph-search characterisations. Overall, the results offer both theoretical structure theorems and practical, scalable algorithms for approximating treebandwidth while highlighting intrinsic parameterised complexity barriers for exact computation.

Abstract

We obtain structure theorems for graphs excluding a fan (a path with a universal vertex) or a dipole () as a topological minor. The corresponding decompositions can be computed in FPT linear time. This is motivated by the study of a graph parameter we call treebandwidth which extends the graph parameter bandwidth by replacing the linear layout by a rooted tree such that neighbours in the graph are in ancestor-descendant relation in the tree. We deduce an approximation algorithm for treebandwidth running in FPT linear time from our structure theorems. We complement this result with a precise characterisation of the parameterised complexity of computing the parameter exactly.

Paper Structure

This paper contains 15 sections, 7 theorems, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. Bandwidth and complexity.
  3. Structure of bounded bandwidth graphs.
  4. Topological bandwidth.
  5. Tree-like generalisations of bandwidth.
  6. Metric flavoured variants.
  7. Treebandwidth.
  8. Treebandwidth and tree-partition-width.
  9. Graphs excluding a $k$-fan.
  10. Preliminaries
  11. Usual graph notations and terminology
  12. SPQR trees
  13. Tree decompositions
  14. Using SPQR trees for the easy cases of the gem and planar graphs
  15. Forbidding a gem

Key Result

Proposition 1

Given a fixed graph $G$, the following statements are equivalent:

Figures (4)

  • Figure 1: The hierarchy of subclasses of chordal graphs (on the left) and the corresponding hierarchy of graph parameters (on the right) obtained by taking the maximum clique minimised over completions into the class.
  • Figure 2: Table of the parameters corresponding to several variants of searching games.
  • Figure 3: Extending a tree-layout to a subdivision.
  • Figure 4: The minimal obstructions to tree-partition-width: the $k$-wall, the $k$-fan, the $k$-multiple of $k$-star, the $k$-multiple of $k$-path. Only the first two are obstructions to treebandwidth.

Theorems & Definitions (7)

  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Theorem 8