Abstract
We show that every graph with pathwidth strictly less than a that contains no path on \(2^b\) vertices as a subgraph has treedepth at most 10ab. The bound is best possible up to a constant factor.
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Notes
In this paper, we are concerned only about non-induced paths.
In [3], the bound is stated in the special case \(t=h=b\), but the proof works in general.
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Acknowledgements
This research was carried out at the “Sparsity in Algorithms, Combinatorics, and Logic” workshop held in Dagstuhl in September 2021. We thank the organizers and the workshop participants for creating a productive working atmosphere. In particular, we thank Marthe Bonamy, Marcin Briański, Kord Eickmeyer, Wojciech Nadara, Ben Rossman, Blair D. Sullivan, and Alexandre Vigny for initial discussions on the topic of the paper. We also thank an anonymous reviewer for noticing an error in a previous version of the paper.
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M. Hatzel was supported by the Federal Ministry of Education and Research (BMBF) and by a fellowship within the IFI programme of the German Academic Exchange Service (DAAD). G. Joret is supported by a CDR grant from the Belgian National Fund for Scientific Research (FNRS), a PDR grant from FNRS, and by the Wallonia Brussels International (WBI) agency. P. Micek is supported by the National Science Center of Poland under grant no. 2018/31/G/ST1/03718. B. Walczak is partially supported by the National Science Center of Poland under grant no. 2019/34/E/ST6/00443. The research of M. Pilipczuk is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme Grant Agreement 714704.
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Hatzel, M., Joret, G., Micek, P. et al. Tight Bound on Treedepth in Terms of Pathwidth and Longest Path. Combinatorica 44, 417–427 (2024). https://doi.org/10.1007/s00493-023-00077-w
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DOI: https://doi.org/10.1007/s00493-023-00077-w