We say a class $\mathcal{C}$ of graphs is clean if for every positive integer t there exists a positive integer $w(t)$ such that every graph in $\mathcal{C}$ with treewidth more than $w(t)$ contains an induced subgraph isomorphic to one of the following: the complete graph $K_t$, the complete bipartite graph $K_{t,t}$, a subdivision of the $(t\times t)$-wall or the line graph of a subdivision of the $(t\times t)$-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Weißauer) to prove that the class of all H-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph H) is clean if and only if H is a forest whose components are subdivided stars.

Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest H as above, we show that forbidding certain connected graphs containing H as an induced subgraph (rather than H itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer η, a complete description of unavoidable connected induced subgraphs of a connected graph G containing η vertices from a suitably large given set of vertices in G. This is of independent interest, and will be used in subsequent papers in this series.

我们说一个类$\mathcal{C}$如果对于每个正整数t都存在一个正整数，则图是干净的$w（t）$使得每个图在$\mathcal{C}$树宽大于$w（t）$包含与以下之一同构的导出子图：完整图$K_t$，完整的二分图$K_{t,t}$, 的一个细分$（t\times t）$-墙或细分的线图$（t\times t）$-墙。在本文中，我们采用 Lozin 和 Razgon 的方法（基于 Weißauer 的早期思想）来证明所有H 无图（即，没有与固定图H同构的诱导子图的图）的类是干净的当且仅当H是一个森林，其组成部分是细分的恒星。

他们的方法很容易应用来产生上述特征。然而，我们的主要结果更加强大：对于上面的每个森林H，我们表明禁止某些包含H作为诱导子图（而不是H本身）的连通图足以获得一类干净的图。沿着后者强化的证明，我们建立在戴维斯的结果的基础上，并为每个正整数η生成连通图G的不可避免的连通诱导子图的完整描述，其中包含来自G中适当大的给定顶点集的η顶点。这是独立的兴趣，并将在本系列的后续论文中使用。