Abrishami, TaraAlecu, BogdanChudnovsky, MariaHajebi, SepehrSpirkl, Sophie2023-11-212023-11-212024-01https://doi.org/10.1016/j.jctb.2023.10.008http://hdl.handle.net/10012/20106The final publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2023.10.008. © 2024. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/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 Kt, the complete bipartite graph Kt,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 verticenAttribution-NonCommercial-NoDerivatives 4.0 Internationalinduced subgraphtree decompositiontreewidthArticle