Induced subgraphs and tree decompositions XIV. Non-adjacent neighbours in a hole

TL;DR

The paper advances the understanding of how induced subgraph obstructions govern treewidth by focusing on clock-free graphs. It introduces the central-bag technique, employs a diamond-free reduction, and leverages three-path configurations (prisms, pyramids, thetas) along with paws and seagulls to locate robust cutsets, culminating in a lifting scheme that converts local separator bounds in a reduced bag to global bounds in the original graph. The main contribution is a proof that clock-free graphs are clean: sufficiently large treewidth guarantees the presence of a large basic obstruction among $K_{t+1}$, $K_{t,t}$, subdivisions of a $(t imes t)$-wall, or the line graph of a subdivision of a $(t imes t)$-wall, thereby connecting induced-subgraph structure to treewidth. The work intricately combines central-bag machinery, 2-clique cutset analysis, and marker-path techniques to control external attachments and derive global treewidth consequences, with potential implications for graph-minor theory and structural graph theory.

Abstract

A clock is a graph consisting of an induced cycle $C$ and a vertex not in $C$ with at least two non-adjacent neighbours in $C$. We show that every clock-free graph of large treewidth contains a "basic obstruction" of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall.

Figures (9)

• Figure 1: The $4$-basic obstructions
• Figure 2: A graph from the Pohoata-Davies construction
• Figure 3: In the proof of Theorem \ref{['thm:paw']}, the choice of $b = b(Q_i)$ and a path with $l(Q_i)$ vertices of the form $P_1(Q_i) \cup \{b\}$ in the case that $Q_i$ is a pyramid ($i=1$) or $Q_i$ is a prism $(i=2)$.
• Figure 4: In the proof of Theorem \ref{['thm:paw']}, the sets $A, Y$ and $Z$ in the case that $Q_i$ is a pyramid ($i=1$) or $Q_i$ is a prism $(i=2)$. Dashed lines represent paths of arbitrary length.
• Figure 5: Proof of Lemma \ref{['lem:seagull']}. Dashed lines represent paths of arbitrary length.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2: Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl, Vušković abrishami2022induced
• Conjecture 1.3
• Lemma 2.1
• proof
• Lemma 3.1: Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl, Vušković abrishami2022induced
• Lemma 3.2: Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl, Vušković abrishami2022induced
• Theorem 3.3
• Lemma 3.4
• proof