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Induced subgraphs and tree decompositions VIII. Excluding a forest in (theta, prism)-free graphs

Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl

TL;DR

The paper characterizes induced-subgraph obstructions in $(\theta, \mathrm{prism})$-free graphs with large treewidth, showing that forests are exactly the graphs that modulate this class. It develops a structural framework built around pyramid-induced strip-structures and jewels to decompose the graph around a trapped apex, then leverages a connectivity/seed approach to bound treewidth. Using a block-to-tree methodology and results of Kierstead–Penrice, it proves that for any tree $F$ and any $t$, graphs in $\mathcal{C}_t(F)$ have bounded treewidth, i.e., there exists $\tau(F,t)$ with $\mathrm{tw}(G) \le \tau(F,t)$. Consequently forests modulate the class of $(\theta, \mathrm{prism})$-free graphs, extending modulation results known for even-hole-free graphs to this broader setting and advancing understanding of induced-subgraph obstructions in high-treewidth graphs.

Abstract

Given a graph $H$, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to $H$, if and only if $H$ is a forest.

Induced subgraphs and tree decompositions VIII. Excluding a forest in (theta, prism)-free graphs

TL;DR

The paper characterizes induced-subgraph obstructions in -free graphs with large treewidth, showing that forests are exactly the graphs that modulate this class. It develops a structural framework built around pyramid-induced strip-structures and jewels to decompose the graph around a trapped apex, then leverages a connectivity/seed approach to bound treewidth. Using a block-to-tree methodology and results of Kierstead–Penrice, it proves that for any tree and any , graphs in have bounded treewidth, i.e., there exists with . Consequently forests modulate the class of -free graphs, extending modulation results known for even-hole-free graphs to this broader setting and advancing understanding of induced-subgraph obstructions in high-treewidth graphs.

Abstract

Given a graph , we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to , if and only if is a forest.
Paper Structure (7 sections, 29 theorems, 10 equations, 2 figures)

This paper contains 7 sections, 29 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and results from the literature
  3. Jumps and jewels on pyramids with trapped apices
  4. Strip structures with an ornament of jewels
  5. Jewels under the loupe
  6. Strip structures and connectivity
  7. From blocks to trees

Key Result

Theorem 1.1

For every integer $t\geq 1$ there exists $w=w(t)\geq 1$ such that every graph of treewidth more than $w$ contains a subdivision of $W_{t \times t}$ as a subgraph.

Figures (2)

  • Figure 1: $W_{5 \times 5}$
  • Figure 2: Theta, pyramid and prism. The dashed lines represent paths of length at least one.

Theorems & Definitions (43)

  • Theorem 1.1: Robertson and Seymour RS-GMV
  • Theorem 1.2: Sintiari and Trotignon layered-wheels
  • Theorem 1.3: Sintiari and Trotignon layered-wheels
  • Theorem 1.4: Korhonen Korhonen
  • Theorem 1.5: Abrishami, Alecu, Chudnovsky, Hajebi and Spirkl twvii
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2.1: Menger Menger
  • Theorem 2.2: Erde and Weiß auer tighttw, see also Grohe
  • ...and 33 more