Induced subgraphs and tree decompositions XII. Grid theorem for pinched graphs

TL;DR

The paper proves a grid-type theorem for induced subgraphs in the class of c-pinched graphs, identifying the exact obstructions that force large treewidth: large complete graphs, large complete bipartite graphs, subdivisions of walls, line graphs of subdivisions of walls, and expansions of Pohoata-Davies constructions. It develops a framework of constellations, alignments, and arrays to translate high local connectivity into a PD_s–style induced obstruction, and shows how to convert a strong block into a plain constellation and then into an array. The main result extends to (c,h)-pinched graphs, giving a complete obstruction description for large treewidth in this broader setting, and it connects to prior grid-type results by generalizing non basic obstructions beyond the basic four. The techniques combine Ramsey theory, patch and bundle arguments, and a two-step approach that first extracts an approximate obstruction (a constellation) and then refines it into an exact induced subgraph obstruction.

Abstract

Given an integer $c\in \mathbb{N}$, we say a graph $G$ is $c$-pinched if $G$ does not contain an induced subgraph consisting of $c$ cycles, all going through a single common vertex and otherwise pairwise disjoint and with no edges between them. What can be said about the structure of $c$-pinched graphs? For instance, $1$-pinched graphs are exactly graphs of treewidth $1$. However, bounded treewidth for $c>1$ is immediately seen to be a false hope because complete graphs, complete bipartite graphs, subdivided walls and line graphs of subdivided walls are all examples of $2$-pinched graphs with arbitrarily large treewidth. There is even a fifth obstruction for larger values of $c$, discovered by Pohoata and later independently by Davies, consisting of $3$-pinched graphs with unbounded treewidth and no large induced subgraph isomorphic to any of the first four obstructions. We fuse the above five examples into a grid-type theorem fully describing the unavoidable induced subgraphs of pinched graphs with large treewidth. More precisely, we prove that for every integer $c\in \mathbb{N}$, a $c$-pinched graph $G$ has large treewidth if and only if $G$ contains one of the following as an induced subgraph: a large complete graph, a large complete bipartite graph, a subdivision of a large wall, the line-graph of a subdivision of a large wall, or a large graph from the Pohoata-Davies construction. Our main result also generalizes to an extension of pinched graphs where the lengths of excluded cycles are lower-bounded.

Figures (8)

• Figure 1: The $6$-by-$6$ square grid (left) and the $6$-by-$6$ wall $W_{6\times 6}$ (right).
• Figure 2: The $4$-basic obstructions.
• Figure 3: The graph $PD_6$ (left) and an expansion of $PD_6$ (right).
• Figure 4: Top: a $(5,1)$-constellation $\mathfrak{c}$ with $S_\mathfrak{c}=\{x_1,x_2,x_3,x_4,x_5\}$ and $L_{\mathfrak{c}}=L$. Note that $\mathfrak{c}$ is $3$-meager (with $u$ being the only vertex in $L$ with three neighbors in $S_{\mathfrak{c}}$) and $6$-hollow (with the $x_3$-gap $v\hbox{-} L\hbox{-} w$ of length five being the longest). Bottom: a $5$-alignment.
• Figure 5: An induced subgraph of $W_{4\times 4}$ isomorphic to a subdivision of $K_4$ with exactly one unsubdivided edge.

Theorems & Definitions (26)

• Theorem 1.1: Robertson and Seymour RS-GMV
• Theorem 1.2: Korhonen Korhonen
• Theorem 1.3: Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl twvii
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Abrishami, Alecu, Chudnovsky, Hajebi, Spirkl twvii
• Theorem 2.2: Dvořák, see Theorem 6 in dvorak; Lozin and Razgon, see Theorem 3 in lozin
• Theorem 3.1
• proof
• Theorem 3.2