Seymour’s Conjecture on 2-Connected Graphs of Large Pathwidth


We prove a conjecture of Seymour (1993) stating that for every apex-forest H1 and out-erplanar graph H2 there is an integer p such that every 2-connected graph of pathwidth at least p contains H1 or H2 as a minor. An independent proof was recently obtained by Dang and Thomas [3].

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We would like to thank Robin Thomas for informing us of the PhD thesis of Thanh N. Dang [2]. We are also grateful to an anonymous referee for several helpful remarks and suggestions.

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Correspondence to David R. Wood.

Additional information

G. Joret acknowledges support from the Australian Research Council and from an Action de Recherche Concertée grant from the Wallonia-Brussels Federation in Belgium. P. Micek is partially supported by a Polish National Science Center grant (SONATA BIS 5; UMO-2015/18/E/ST6/00299). T. Huynh is supported by ERC Consolidator Grant 615640-ForEFront. T. Huynh, G. Joret, and P. Micek also acknowledge support from a joint grant funded by the Belgian National Fund for Scientific Research (F.R.S.-FNRS) and the Polish Academy of Sciences (PAN). Research of D. R. Wood is supported by the Australian Research Council.

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Huynh, T., Joret, G., Micek, P. et al. Seymour’s Conjecture on 2-Connected Graphs of Large Pathwidth. Combinatorica 40, 839–868 (2020).

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Mathematics Subject Classification (2010)

  • 05C83