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Induced subgraphs and tree decompositions XIX. Thetas and forests

Maria Chudnovsky, Julien Codsi, Sepehr Hajebi, Sophie Spirkl

TL;DR

The paper tackles the problem of bounding the treewidth of theta-free graphs under the dual restrictions of excluding line graphs of subdivisions of walls and forbidding a forest $H$ as an induced subgraph. It introduces a separability framework and proves that theta-free $(H,K_t)$-free graphs are $t^{d}$-separable for suitable $d=d(H)$, enabling a reduction to low-separability that leverages existing induced-minor results to obtain a polynomial bound on treewidth in terms of the clique number. By combining Ramsey-type arguments, constellation lemmas, and induced-minor machinery, it derives a bound of the form $\mathrm{tw}(G) \le \omega(G)^{d(H,r)}$ for graphs avoiding line-graph subdivisions of $W_{r\times r}$, with $H$ a forest. This leads to structural consequences, including equivalences among conditions for bounded treewidth, poly-logarithmic tree-independence numbers, and quasi-polynomial-time solvability for problems like Maximum Weight Independent Set within these classes, enriching the induced-subgraph analogue of the grid theorem.

Abstract

Let $H$ be a graph and let $\mathcal{C}$ be a hereditary class of theta-free graphs such that $H\notin \mathcal{C}$. We prove that if (a) $H$ is a forest; and (b) $\mathcal{C}$ excludes the line graphs of all subdivisions of some wall, then the treewidth of every graph in $\mathcal{C}$ is at most a polynomial function of its clique number. This is best possible in that both (a) and (b) are necessary for the existence of $any$ function with the above property.

Induced subgraphs and tree decompositions XIX. Thetas and forests

TL;DR

The paper tackles the problem of bounding the treewidth of theta-free graphs under the dual restrictions of excluding line graphs of subdivisions of walls and forbidding a forest as an induced subgraph. It introduces a separability framework and proves that theta-free -free graphs are -separable for suitable , enabling a reduction to low-separability that leverages existing induced-minor results to obtain a polynomial bound on treewidth in terms of the clique number. By combining Ramsey-type arguments, constellation lemmas, and induced-minor machinery, it derives a bound of the form for graphs avoiding line-graph subdivisions of , with a forest. This leads to structural consequences, including equivalences among conditions for bounded treewidth, poly-logarithmic tree-independence numbers, and quasi-polynomial-time solvability for problems like Maximum Weight Independent Set within these classes, enriching the induced-subgraph analogue of the grid theorem.

Abstract

Let be a graph and let be a hereditary class of theta-free graphs such that . We prove that if (a) is a forest; and (b) excludes the line graphs of all subdivisions of some wall, then the treewidth of every graph in is at most a polynomial function of its clique number. This is best possible in that both (a) and (b) are necessary for the existence of function with the above property.

Paper Structure

This paper contains 3 sections, 18 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reducing to low separability
  3. Growing a tree

Key Result

Theorem 1.1

For every $t\in \mathbb{N}$, graphs with no minor (or equivalently, no subgraph) isomorphic to any subdivision of $W_{t\times t}$ have bounded treewidth.

Figures (1)

  • Figure 1: A subdivision of $W_{4\times 4}$ (left), its line graph (middle), and a theta in $W_{3\times 3}$ (right).

Theorems & Definitions (22)

  • Theorem 1.1: Robertson and Seymour GMV
  • Theorem 1.2: Sintiari and Trotignon layeredwheels
  • Corollary 1.3
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5: Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl tw8
  • Corollary 1.6
  • Theorem 2.0
  • Theorem 2.1: Hajebi; Theorem 3.2 in polypw for $\kappa=2$
  • Theorem 2.2: Chudnovsky, Hajebi, Spirkl; Theorem 1.2 in tw16
  • ...and 12 more