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Induced subgraphs and tree decompositions XVII. Anticomplete sets of large treewidth

Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl

TL;DR

The paper investigates when graphs of sufficiently large treewidth must contain two anticomplete sets whose induced subgraphs both have large treewidth, or else admit specific obstructions. It develops a robust framework of models (μ-models and (ρ,σ)-models) and leverages Ramsey theory to move from half-induced to fully induced configurations, ultimately establishing a comprehensive dichotomy. The main contributions include a sequence of structural results: complete models, half-induced models, and a bridge from half-induced to induced subgraphs, culminating in a global theorem that characterizes obstructions to large induced treewidth and identifies interrupted s-constellations as essential counterexamples. The work extends the spirit of the Grid Theorem to induced subgraphs and connects induced-minor obstructions to induced-subgraph structure, with significant use of alignment techniques and advanced Ramsey-type tools.

Abstract

Two sets $X, Y$ of vertices in a graph $G$ are "anticomplete" if $X\cap Y=\varnothing$ and there is no edge in $G$ with an end in $X$ and an end in $Y$. We prove that every graph $G$ of sufficiently large treewidth contains two anticomplete sets of vertices each inducing a subgraph of large treewidth unless $G$ contains, as an induced subgraph, a highly structured graph of large treewidth that is an obvious counterexample to this statement. These are: complete graphs, complete bipartite graphs and "interrupted $s$-constellations." The latter is a slightly adjusted version of a well-known construction by Bonamy et al.

Induced subgraphs and tree decompositions XVII. Anticomplete sets of large treewidth

TL;DR

The paper investigates when graphs of sufficiently large treewidth must contain two anticomplete sets whose induced subgraphs both have large treewidth, or else admit specific obstructions. It develops a robust framework of models (μ-models and (ρ,σ)-models) and leverages Ramsey theory to move from half-induced to fully induced configurations, ultimately establishing a comprehensive dichotomy. The main contributions include a sequence of structural results: complete models, half-induced models, and a bridge from half-induced to induced subgraphs, culminating in a global theorem that characterizes obstructions to large induced treewidth and identifies interrupted s-constellations as essential counterexamples. The work extends the spirit of the Grid Theorem to induced subgraphs and connects induced-minor obstructions to induced-subgraph structure, with significant use of alignment techniques and advanced Ramsey-type tools.

Abstract

Two sets of vertices in a graph are "anticomplete" if and there is no edge in with an end in and an end in . We prove that every graph of sufficiently large treewidth contains two anticomplete sets of vertices each inducing a subgraph of large treewidth unless contains, as an induced subgraph, a highly structured graph of large treewidth that is an obvious counterexample to this statement. These are: complete graphs, complete bipartite graphs and "interrupted -constellations." The latter is a slightly adjusted version of a well-known construction by Bonamy et al.

Paper Structure

This paper contains 7 sections, 22 theorems, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Models, strong blocks and an outline of the proof
  3. Complete models
  4. Half-induced models
  5. From half induced to induced
  6. From induced minors to induced subgraphs
  7. Acknowledgements

Key Result

Theorem 1.1

For every integer $r\in \mathbb{N}$, there is a constant $f_{thm:wallminor}=f_{thm:wallminor}(r)\in \mathbb{N}$ such that every graph $G$ with $\mathop{\mathrm{tw}}\nolimits(G) \geq f_{thm:wallminor}$ has a subgraph isomorphic to a subdivision of $W_{r\times r}$.

Figures (2)

  • Figure 1: Examples of graphs with large treewidth: $K_{4,4}$ (top left), $K_5$ (bottom left), a subdivision of the $W_{4\times 4}$ and the line graph of a subdivision of the $W_{4\times 4}$ (right).
  • Figure 2: An ample interrupted $4$-constellation.

Theorems & Definitions (32)

  • Theorem 1.1: Robertson and Seymour GMV
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2: Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl tw7
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3: Ramsey multiramsey
  • Theorem 3.4: Graham, Rothschild and Spencer productramsey
  • Theorem 3.5: Ramsey multiramsey
  • ...and 22 more