Induced subgraphs and tree decompositions VII. Basic obstructions in H-free graphs.
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Date
2024-01
Authors
Abrishami, Tara
Alecu, Bogdan
Chudnovsky, Maria
Hajebi, Sepehr
Spirkl, Sophie
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
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 × 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 vertic
Description
The 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/
Keywords
induced subgraph, tree decomposition, treewidth