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

JOURNAL OF COMBINATORIAL THEORY SERIES B（2024）

Abstract

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 x t)wall or the line graph of a subdivision of the (t x 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

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Key words

Induced subgraph,Tree decomposition,Treewidth

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