Treewidth versus clique number: induced minors

TL;DR

This paper establishes a fundamental equivalence between the boundedness of treewidth and clique number in hereditary graph classes and their induced minors, providing a key insight into graph structure theory.

Contribution

It proves that a hereditary class is $( ext{tw}, ext{ω})$-bounded if and only if its induced minors are also $( ext{tw}, ext{ω})$-bounded, linking two important graph properties.

Findings

Hereditary classes with bounded treewidth and clique number are characterized by their induced minors.

Induced minors preserve the $( ext{tw}, ext{ω})$-boundedness property.

The result offers a new perspective on graph minor theory and hereditary class characterization.

Abstract

We prove that a hereditary class of graphs is $(\mathsf{tw}, \omega)$-bounded if and only if the induced minors of the graphs from the class form a $(\mathsf{tw}, \omega)$-bounded class.