Treewidth versus clique number. IV. Tree-independence number of graphs excluding an induced star

TL;DR

This paper investigates the relationship between treewidth, clique number, and the newly introduced tree-independence number in graphs excluding an induced star, providing theoretical results and algorithms for specific graph classes.

Contribution

It proves the conjecture that $(tw, ext{omega})$-boundedness is equivalent to bounded tree-independence number for graphs excluding an induced star, and characterizes the tree-independence number for several graph classes.

Findings

Proved the conjecture for graphs excluding an induced star.

Determined the tree-independence number for line graphs of complete graphs.

Developed a linear-time algorithm for computing the tree-independence number of $P_4$-free graphs.

Abstract

Many recent works address the question of characterizing induced obstructions to bounded treewidth. In 2022, Lozin and Razgon completely answered this question for graph classes defined by finitely many forbidden induced subgraphs. Their result also implies a characterization of graph classes defined by finitely many forbidden induced subgraphs that are $(tw,\omega)$-bounded, that is, treewidth can only be large due to the presence of a large clique. This condition is known to be satisfied for any graph class with bounded 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. conjectured that $(tw,\omega)$-boundedness is actually equivalent to bounded tree-independence number. We address this conjecture in the context of graph classes defined by finitely many forbidden induced subgraphs and prove it for the case of graph classes excluding an induced star. We also prove it for subclasses of the class of line graphs, determine the exact values of the tree-independence numbers of line graphs of complete graphs and line graphs of complete bipartite graphs, and characterize the tree-independence number of $P_4$-free graphs, which implies a linear-time algorithm for its computation. Applying the algorithmic framework provided in a previous paper of the series leads to polynomial-time algorithms for the Maximum Weight Independent Set problem in an infinite family of graph classes.