Every Graph is Essential to Large Treewidth

TL;DR

This paper constructs hereditary graph classes with unbounded treewidth where excluding any fixed graph of bounded treewidth results in graphs of bounded treewidth, challenging existing conjectures and expanding the understanding of graph structure.

Contribution

It introduces hereditary graph classes with unbounded treewidth that contain only bounded treewidth graphs when excluding any fixed bounded treewidth graph, providing new counterexamples.

Findings

Hereditary classes with unbounded treewidth exist where excluding any fixed bounded treewidth graph yields bounded treewidth graphs.

The construction uses layered wheels and introduces a framework of abstract layered wheels.

This work refutes several conjectures about unavoidable induced subgraphs in classes of unbounded 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.