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

TL;DR

This paper advances the understanding of when $(tw,\omega)$-boundedness coincides with bounded tree-independence number by establishing a suite of equivalences for specific graph classes, notably the complements of line graphs, and by providing new results for star-excluding and finitely forbidden subgraph families. It leverages splitness theory, induced-minor analysis, and structural decompositions to relate $(tw,\omega)$-boundedness to bounded $\mathsf{tree}\text{-}\alpha$, $K_{s,s}$-freeness, and subgraph-free conditions, yielding concrete bounds and exact values in several cases. Key contributions include a complete equivalence framework for $\overline{L(\mathcal G)}$, simpler proofs for the $K_{1,t}$-free $\mathcal{O}_k$-free regime, precise results for $(P_3+P_1)$-free graphs linking $\mathsf{tree}\text{-}\alpha$ to $\mathsf{ibn}$, and a bound of $3$ on the tree-clique-cover number for $\{P_4+P_1,C_4\}$-free graphs via induced-minor-free results. These findings sharpen the boundary between $(tw,\omega)$-boundedness and tree-independence for dense graph classes and inform algorithmic implications for clique and independent-set problems in these families.

Abstract

We continue the study of $(tw,ω)$-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, Milanič, and Štorgel in 2024. Dallard et al. showed that bounded tree-independence number is sufficient for $(tw,ω)$-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 $(tw,ω)$-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 $(tw, ω)$-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.

Theorems & Definitions (50)

• Conjecture 1.1: Dallard et al. dallard2023treewidth
• Theorem 1.2: Dallard et al. dallard2024treewidth
• Corollary 1.3
• Conjecture 1.4: Conjecture 1.2 in dallard2024treewidth
• Conjecture 1.5: Conjecture 1.3 in dallard2024treewidth
• Conjecture 1.6
• Theorem 1.7
• Theorem 1.8: Ahn et al. ahn2024coarseerdhosposatheorem
• Theorem 1.9
• Theorem 1.10