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：近期众多研究聚焦于刻画导致有界树宽的诱导障碍。2022年，Lozin与Razgon完全解答了关于有限个禁止诱导子图定义的图类的问题。他们的结果也刻画了由有限个禁止诱导子图定义的且具有$(tw,\omega)$-有界性的图类，即树宽仅因大团的存在而变大。已知该条件适用于任何具有有界树-独立数的图类——该图参数由Yolov（2018年）以及Dallard、Milani\v{c}与\v{S}torgel（2024年）独立提出。Dallard等人猜想$(tw,\omega)$-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究此猜想，并证明了不含诱导星的图类情形。同时，我们亦对线图子类证明了该猜想，确定了完全图与完全二分图的线图的树-独立数精确值，并刻画了$P_4$-自由图的树-独立数（由此可在线性时间内计算）。应用本系列先前论文提供的算法框架，可为无限族图类中的最大权独立集问题提供多项式时间算法。