Dabrowski, KK orcid.org/0000-0001-9515-6945, Dross, F, Jeong, J et al. (4 more authors) (2022) Tree Pivot-Minors and Linear Rank-Width. SIAM Journal on Discrete Mathematics, 35 (4). pp. 2922-2945. ISSN 0895-4801
Abstract
Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour [J. Combin. Theory Ser. B, 35 (1983), pp. 39--61] proved that for every tree $T$, the class of graphs that do not contain $T$ as a minor has bounded path-width. For the pivot-minor relation, rank-width and linear rank-width take over the role of tree-width and path-width. As such, it is natural to examine if, for every tree $T$, the class of graphs that do not contain $T$ as a pivot-minor has bounded linear rank-width. We first prove that this statement is false whenever $T$ is a tree that is not a caterpillar. We conjecture that the statement is true if $T$ is a caterpillar. We are also able to give partial confirmation of this conjecture by proving for every tree $T$, the class of $T$-pivot-minor-free distance-hereditary graphs has bounded linear rank-width if and only if $T$ is a caterpillar; for every caterpillar $T$ on at most four vertices, the class of $T$-pivot-minor-free graphs has bounded linear rank-width. To prove our second result, we only need to consider $T=P_4$ and $T=K_{1,3}$, but we follow a general strategy: first we show that the class of $T$-pivot-minor-free graphs is contained in some class of $(H_1,H_2)$-free graphs, which we then show to have bounded linear rank-width. In particular, we prove that the class of $(K_3,S_{1,2,2})$-free graphs has bounded linear rank-width, which strengthens a known result that this graph class has bounded rank-width.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021, Society for Industrial and Applied Mathematics. This is an author produced version of an article accepted for publication in SIAM Journal on Discrete Mathematics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | tree; pivot-minor; linear rank-width |
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: | 01 Sep 2021 11:16 |
Last Modified: | 06 Dec 2024 17:01 |
Status: | Published |
Publisher: | Society for Industrial and Applied Mathematics |
Identification Number: | 10.1137/21M1402339 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:177573 |