Every Graph is Essential to Large Treewidth

2025-02-20

AI Summary

This paper addresses the central conjecture: “Do unbounded-treewidth hereditary graph classes necessarily contain every fixed graph $H$ as an induced subgraph?” The authors construct a family $mathcal{C}_H$ of hereditary graphs with unbounded treewidth, and prove that excluding any fixed $H$ as an induced subgraph from $mathcal{C}_H$ yields a subclass of bounded treewidth. This constitutes the first systematic refutation of several long-standing conjectures asserting the inevitability of arbitrary induced subgraphs in unbounded-treewidth hereditary classes. Technically, the work introduces an abstract hierarchical wheel framework—unifying and generalizing prior hierarchical wheel lemmas—and leverages it to build the first universal family of counterexamples. Combining hereditary closure, treewidth analysis, and induced-subgraph exclusion techniques, the paper reveals a fundamental mechanism: graphs $H$ of low treewidth can be “locally eliminated” without compromising the unboundedness of treewidth in the overall class.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Refutes conjectures on unavoidable induced subgraphs.
• Introduces hereditary graph classes with unbounded treewidth.
• Develops framework for abstract layered wheels.

Innovation

• Hereditary graph class
• Layered wheels variant
• Abstract layered wheels framework