Every Graph is Essential to Large Treewidth

TL;DR

This work shows that no fixed induced subgraph characterizes large treewidth within every unbounded-treewidth class: for any graph $H$ there exists a hereditary weakly sparse class with unbounded treewidth whose $H$-free graphs have bounded treewidth. The authors develop an abstract layered-wheel framework and a concrete upward-restricted layered wheel $W_t$ to construct such classes, proving that $H$-free subfamilies have bounded treewidth when $\text{tw}(H)\le t$ and that the finite induced subgraphs of $W_t$ have bounded treewidth in these cases. They also obtain a triangle-free variant and show how these constructions refute several conjectures about unavoidable induced subgraphs, while connecting to twin-width and separation-number techniques. The results have broad implications for understanding the limits of structural graph theory in forcing complex substructures and for designing counterexamples to proposed universal properties of graphs with large treewidth.

Abstract

We show that for every graph $H$, there is a hereditary weakly sparse graph class $\mathcal C_H$ of unbounded treewidth such that the $H$-free (i.e., excluding $H$ as an induced subgraph) graphs of $\mathcal C_H$ have bounded treewidth. This refutes several conjectures and critically thwarts the quest for the unavoidable induced subgraphs in classes of unbounded treewidth, a wished-for counterpart of the Grid Minor theorem. We actually show a stronger result: For every positive integer $t$, there is a hereditary graph class $\mathcal C_t$ of unbounded treewidth such that for any graph $H$ of treewidth at most $t$, the $H$-free graphs of $\mathcal C_t$ have bounded treewidth. Our construction is a variant of so-called layered wheels. We also introduce a framework of abstract layered wheels, based on their most salient properties. In particular, we streamline and extend key lemmas previously shown on individual layered wheels. We believe that this should greatly help develop this topic, which appears to be a very strong yet underexploited source of counterexamples.

Figures (6)

• Figure 1: Possible first five layers of a proper, neat, 3-bounded layered wheel $G$ with rooted tree $T$. The edges of $T$ are dotted, the layer edges of $G$ are in blue, and its other edges are in black (with white dots if they are also edges of $T$).
• Figure 2: The $5 \times 5$ grid (left) and the $5 \times 5$ wall (right).
• Figure 3: A chordal trigraph $H$ and a tree representation $T$ of $H$.
• Figure 4: First three layers of $T_1, G_1, W_1$.
• Figure 5: First three layers of $T_t, G_t, W_t$ when $t \geqslant 2$.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1: Hajebi, Conjecture 1.15 of Hajebi24, previously refuted in chudnovsky2024treewidthmaximumcliques
• Conjecture 2: Hajebi, Conjecture 1.14 of Hajebi24, previously refuted in chudnovsky2024treewidthmaximumcliques
• Conjecture 3: Trotignon
• Conjecture 4
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6