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Induced subgraphs and tree decompositions V. One neighbor in a hole

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

TL;DR

The paper identifies a refined obstruction framework for bounded treewidth by focusing on holes with external neighbors and related structures. It introduces the helpful family ${\mathcal T}_2$ (holes, prisms, pyramids) and develops a central-bag approach grounded in balanced separators and two-clique separations to handle heavy seagulls. By reducing to triangle-free cases and proving a robust $t=2$ result, it derives a general bound: for every $t>0$, graphs free of ${\mathcal F}_t$ with large treewidth contain an induced subdivision of $W_{t\times t}$. This advances understanding of induced-obstruction conditions that force large treewidth and provides a blueprint for structural decomposition via central bags and separator-based arguments.

Abstract

What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the "basic treewidth obstructions"). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so-called "layered wheels," contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.

Induced subgraphs and tree decompositions V. One neighbor in a hole

TL;DR

The paper identifies a refined obstruction framework for bounded treewidth by focusing on holes with external neighbors and related structures. It introduces the helpful family (holes, prisms, pyramids) and develops a central-bag approach grounded in balanced separators and two-clique separations to handle heavy seagulls. By reducing to triangle-free cases and proving a robust result, it derives a general bound: for every , graphs free of with large treewidth contain an induced subdivision of . This advances understanding of induced-obstruction conditions that force large treewidth and provides a blueprint for structural decomposition via central bags and separator-based arguments.

Abstract

What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the "basic treewidth obstructions"). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so-called "layered wheels," contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.
Paper Structure (10 sections, 27 theorems, 7 equations, 2 figures)

This paper contains 10 sections, 27 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Organization of the paper
  4. Separations
  5. Reducing to the triangle-free case
  6. Balanced separators and central bags
  7. Two-clique-separations
  8. Heavy seagulls
  9. Proof of Theorem \ref{['t=2']}
  10. Putting everything together

Key Result

Theorem 1.1

There is a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that every graph of treewidth at least $f(k)$ contains a subdivision of $W_{k \times k}$ as a subgraph.

Figures (2)

  • Figure 1: $W_{5 \times 5}$
  • Figure 2: A wheel-free graph with large treewidth Davies

Theorems & Definitions (42)

  • Theorem 1.1: RS-GMV
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 3.1
  • ...and 32 more