Every Graph is Essential to Large Treewidth
Bogdan Alecu, \'Edouard Bonnet, Pedro Bureo Villafana, and Nicolas, Trotignon

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 , there is a hereditary weakly sparse graph class of unbounded treewidth such that the -free (i.e., excluding as an induced subgraph) graphs of 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 , there is a hereditary graph class of unbounded treewidth such that for any graph of treewidth at most , the -free graphs of 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…
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Taxonomy
TopicsAdvanced Graph Theory Research
