Aboulker, P, Adler, I orcid.org/0000-0002-9667-9841, Kim, EJ et al. (2 more authors) (2021) On the tree-width of even-hole-free graphs. European Journal of Combinatorics, 98. 103394. ISSN 0195-6698
Abstract
The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, K₄)-free graphs was constructed, that still has unbounded tree-width (Sintiari and Trotignon, 2019). The class has unbounded degree and contains arbitrarily large clique-minors. We ask whether this is necessary.
We prove that for every graph G, if G excludes a fixed graph H as a minor, then G either has small tree-width, or G contains a large wall or the line graph of a large wall as induced subgraph. This can be seen as a strengthening of Robertson and Seymour’s excluded grid theorem for the case of minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. This implies the known result that planar even hole-free graphs have bounded tree-width (Silva et al., 2010).
We conjecture that even-hole-free graphs of bounded degree have bounded tree-width. If true, this would mean that even-hole-freeness is testable in the bounded-degree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the tree-width of the class of (even hole, pyramid)-free graphs of degree at most 4.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 Elsevier Ltd. All rights reserved. This is an author produced version of an article published in European Journal of Combinatorics. Uploaded in accordance with the publisher's self-archiving policy. |
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) |
Depositing User: | Symplectic Publications |
Date Deposited: | 16 Mar 2021 15:07 |
Last Modified: | 12 Jul 2022 00:14 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.ejc.2021.103394 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:172174 |