Treewidth versus clique number. V. Further connections with tree-independence number

TL;DR

This paper explores the relationship between treewidth, clique number, and tree-independence number in graph classes, identifying conditions where bounded tree-independence number characterizes $(tw, ext{}\omega)$-boundedness, especially for classes with forbidden induced subgraphs.

Contribution

It extends understanding of $(tw, ext{ }\omega)$-boundedness by identifying new classes where it is equivalent to bounded tree-independence number, including subclasses with forbidden induced subgraphs.

Findings

Established equivalences for $(tw, ext{ }\omega)$-boundedness in certain graph classes.

Provided characterizations for complements of line graphs.

Proved bounded tree-independence number for graphs excluding a fixed induced star and cycles.

Abstract

We continue the study of $(tw,\omega)$-bounded graph classes, that is, hereditary graph classes in which large treewidth is witnessed by the presence of a large clique, and the relation of this property to boundedness of the tree-independence number, a graph parameter introduced independently by Yolov in 2018 and by Dallard, Milani\v{c}, and \v{S}torgel in 2024. Dallard et al. showed that bounded tree-independence number is sufficient for $(tw,\omega)$-boundedness, and conjectured that the converse holds. While this conjecture has been recently disproved, it is still interesting to determine classes where the conjecture holds; for example, the conjecture is still open for graph classes excluding an induced star, as well as for finitely many forbidden induced subgraphs. In this paper, we identify further families of graph classes where $(tw,\omega)$-boundedness is equivalent to bounded tree-independence number. We settle a number of cases of finitely many forbidden induced subgraphs, obtain several equivalent characterizations of $(tw, \omega)$-boundedness in subclasses of the class of complements of line graphs, and give a short proof of a recent result of Ahn, Gollin, Huynh, and Kwon [SODA 2025] establishing bounded tree-independence number for graphs excluding a fixed induced star and a fixed number of independent cycles.