Induced subgraphs and tree decompositions VII. Basic obstructions in H-free graphs.

We say a class C of graphs is clean if for every positive integer t there exists a positive integer w (t) such that every graph in 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 × t) -wall or the line graph of a subdivision of the (t × 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. [ABSTRACT FROM AUTHOR]