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Treedepth vs Circumference


The circumference of a graph G is the length of a longest cycle in G, or \(+\infty \) if G has no cycle. Birmelé (J Graph Theory 43(1):24–25, 2003) showed that the treewidth of a graph G is at most its circumference minus 1. We strengthen this result for 2-connected graphs as follows: If G is 2-connected, then its treedepth is at most its circumference. The bound is best possible and improves on an earlier quadratic upper bound due to Marshall and Wood (J Graph Theory 79(3):222–232, 2015).

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  1. Birmelé, E.: Tree-width and circumference of graphs. J. Graph Theory 43(1), 24–25 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bodlaender, H., Gilbert, J., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms 18(2), 238–255 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, J., Czerwinski, W., Disser, Y., Feldmann, A. E., Hermelin, D., Nadara, W., Pilipczuk, M., Pilipczuk, M., Sorge, M., Wróblewski, B., Zych-Pawlewicz, A.: Efficient fully dynamic elimination forests with applications to detecting long paths and cycles. In: Marx, D. (ed.) Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, pp. 796–809. SIAM (2021)

  4. Diestel, R.: Graph Theory, 5th edn. Springer, Berllin (2017)

    Book  MATH  Google Scholar 

  5. Dirac, G.A.: Some theorems on abstract graphs. Proc. Lond. Math. Soc. 3(2), 69–81 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  6. Marshall, E.A., Wood, D.R.: Circumference and pathwidth of highly connected graphs. J. Graph Theory 79(3), 222–232 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Nešetřil, J., de Mendez, P.O.: Sparsity. Algorithms and Combinatorics. Graphs, Structures, and Algorithms, vol. 28. Springer, Heidelberg (2012)

  8. Reidl, F., Rossmanith, P., Villaamil, F. S., Sikdar, S.: A faster parameterized algorithm for treedepth. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) Automata, Languages, and Programming—41st International Colloquium, ICALP 2014, Proceedings, Part I. Lecture Notes in Computer Science, vol. 8572, pp. 931–942. Springer, Berlin (2014)

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This research was started at the Structural Graph Theory workshop in Gułtowy (Poland) in June 2022 organized by Andrzej Grzesik, Marcin Pilipczuk, and Marcin Witkowski. We thank the organizers and the other workshop participants for creating a productive working atmosphere. We thank Michał Pilipczuk for pointing out to us the algorithmic application mentioned in the introduction. We are grateful to the two anonymous referees for their helpful comments.

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Correspondence to Gwenaël Joret.

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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. This work is a part of Project BOBR (K. Majewski) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 948057). P. Micek, M. T. Seweryn and M. Briański are supported by the National Science Center of Poland under Grant UMO-2018/31/G/ST1/03718 within the BEETHOVEN program.

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Briański, M., Joret, G., Majewski, K. et al. Treedepth vs Circumference. Combinatorica 43, 659–664 (2023).

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