Abrishami, T., Alecu, B. orcid.org/0000-0002-5515-9145, Chudnovsky, M. et al. (2 more authors) (2024) Induced subgraphs and tree decompositions VII. Basic obstructions in H-free graphs. Journal of Combinatorial Theory, Series B, 164. pp. 443-472. ISSN 0095-8956
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 vertices in G. This is of independent interest, and will be used in subsequent papers in this series.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2023 Elsevier Inc. All rights reserved. This is an author produced version of an article published in Journal of Combinatorial Theory, Series B made available under the CC-BY-NC-ND 4.0 license (http://creativecommons.org/licenses/by-nc-nd/4.0) in accordance with the publisher's self-archiving policy. |
Keywords: | Induced subgraph; Tree decomposition; Treewidth |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Computing (Leeds) > Algorithms & Complexity |
Depositing User: | Symplectic Publications |
Date Deposited: | 25 Apr 2024 14:36 |
Last Modified: | 07 Nov 2024 01:15 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.jctb.2023.10.008 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:211813 |