Tree pivot-minors and linear rank-width

Dabrowski, K.K.; Dross, F.; Jeong, J.; Kante, M.M.; Kwon, O.; Oum, S.; Paulusma, D.

Abstract

Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) 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 “ P4 and T “ K1,3, but we follow a general strategy: first we show that the class of T-pivot-minor-free graphs is contained in some class of pH1, H2q-free graphs, which we then show to have bounded linear rank-width. In particular, we prove that the class of pK3, S1,2,2q-free graphs has bounded linear rank-width, which strengthens a known result that this graph class has bounded rank-width.

Citation

Dabrowski, K., Dross, F., Jeong, J., Kante, M., Kwon, O., Oum, S., & Paulusma, D. (2021). Tree pivot-minors and linear rank-width. SIAM Journal on Discrete Mathematics, 35(4), 2922-2945. https://doi.org/10.1137/21m1402339

• Journal Article Type: Article
• Acceptance Date: Aug 11, 2021
• Online Publication Date: Dec 7, 2021
• Publication Date: 2021
• Deposit Date: Aug 23, 2021
• Publicly Available Date: Aug 24, 2021
• Journal: SIAM Journal on Discrete Mathematics
• Print ISSN: 0895-4801
• Electronic ISSN: 1095-7146
• Publisher: Society for Industrial and Applied Mathematics
• Peer Reviewed: Peer Reviewed
• Volume: 35
• Issue: 4
• Pages: 2922-2945
• DOI: https://doi.org/10.1137/21m1402339
• Public URL: https://durham-repository.worktribe.com/output/1266267