  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 Wei{\ss}auer) 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 $\eta$, a complete description of unavoidable
connected induced subgraphs of a connected graph $G$ containing $\eta$ 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.
