We continue the study of $(\mathrm{tw},\omega)$-bounded graph classes, that is, hereditary graph classes in which the treewidth can only be large due to the presence of a large clique, with the goal of understanding the extent to which this property has useful algorithmic implications for the Independent Set and related problems. In the previous paper of the series [Dallard, Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. II. Tree-independence number], we introduced the tree-independence number, a min-max graph invariant related to tree decompositions. Bounded tree-independence number implies both $(\mathrm{tw},\omega)$-boundedness and the existence of a polynomial-time algorithm for the Maximum Weight Independent Set problem, provided that the input graph is given together with a tree decomposition with bounded independence number. In this paper, we consider six graph containment relations and for each of them characterize the graphs $H$ for which any graph excluding $H$ with respect to the relation admits a tree decomposition with bounded independence number. The induced minor relation is of particular interest: we show that excluding either a $K_5$ minus an edge or the $4$-wheel implies the existence of a tree decomposition in which every bag is a clique plus at most $3$ vertices, while excluding a complete bipartite graph $K_{2,q}$ implies the existence of a tree decomposition with independence number at most $2(q-1)$. Our constructive proofs are obtained using a variety of tools, including $\ell$-refined tree decompositions, SPQR trees, and potential maximal cliques. They imply polynomial-time algorithms for the Independent Set and related problems in an infinite family of graph classes; in particular, the results apply to the class of $1$-perfectly orientable graphs, answering a question of Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius from 2019.

翻译：我们继续研究 $(\mathrm{tw},\omega)$-有界图类，即树宽仅因大团的存在而变大的遗传图类，旨在理解这一性质对独立集及相关问题的算法意义。在本系列的前一篇论文 [Dallard, Milani\v{c} 和 \v{S}torgel，树宽与团数之关系 II：树独立数] 中，我们引入了树独立数——一种与树分解相关的极小极大图不变量。有界树独立数同时蕴含 $(\mathrm{tw},\omega)$-有界性，以及当输入图附带具有有界独立数的树分解时，最大权独立集问题存在多项式时间算法。本文考虑六种图包含关系，并对每种关系刻画了使得所有在该关系下排除 $H$ 的图均存在具有有界独立数的树分解的图 $H$。特别值得关注的是诱导子式关系：我们证明，排除 $K_5$ 减一条边或 $4$-轮图可推导出存在一个树分解，其中每个袋子均为一个团加上至多 $3$ 个顶点；而排除完全二分图 $K_{2,q}$ 可推导出存在一个树分解，其独立数至多为 $2(q-1)$。我们的构造性证明使用了多种工具，包括 $\ell$-精细化树分解、SPQR 树和潜在极大团。这些证明蕴含了无限族图类中独立集及相关问题的多项式时间算法；特别地，结果适用于 $1$-完美可定向图类，回答了 Beisegel、Chudnovsky、Gurvich、Milani\v{c} 和 Servatius 在 2019 年提出的一个问题。