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.

翻译：近年来，许多研究聚焦于刻画导致有界树宽的诱导障碍结构。2022年，Lozin与Razgon完整回答了由有限个禁止诱导子图定义的图类中的这一问题。他们的结果还意味着对有限个禁止诱导子图定义的图类中满足$(tw,\omega)$-有界性（即树宽只能因大团的存在而增大）的图类进行了刻画。已知任何具有有界树独立数的图类均满足该条件——树独立数由Yolov（2018年）与Dallard、Milani\v{c}及\v{S}torgel（2024年）独立提出的图参数。Dallard等人推测$(tw,\omega)$-有界性实际上等价于有界树独立数。我们针对由有限个禁止诱导子图定义的图类探讨这一猜想，并在排除诱导星图的图类中证明了该猜想。我们还对线图类的子类证明了该猜想，确定了完全图的线图与完全二部图的线图的树独立数精确值，并刻画了$P_4$-自由图的树独立数（由此可推导出其线性时间算法）。应用本系列先前论文提供的算法框架，可得到无限族图类中最大权独立集问题的多项式时间算法。