First Order Logic and Twin-Width in Tournaments

Authors Colin Geniet , Stéphan Thomassé

Thumbnail PDF


  • Filesize: 0.77 MB
  • 14 pages

Document Identifiers

Author Details

Colin Geniet
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France
Stéphan Thomassé
  • Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France


The authors would like to thank Édouard Bonnet for stimulating discussions on this topics, and Szymon Toruńczyk for helpful explanations on the notion of restrained classes.

Cite AsGet BibTex

Colin Geniet and Stéphan Thomassé. First Order Logic and Twin-Width in Tournaments. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments T, first-order model checking either is fixed parameter tractable, or is AW[*]-hard. This dichotomy coincides with the fact that T has either bounded or unbounded twin-width, and that the growth of T is either at most exponential or at least factorial. From the model-theoretic point of view, we show that NIP classes of tournaments coincide with bounded twin-width. Twin-width is also characterised by three infinite families of obstructions: T has bounded twin-width if and only if it excludes at least one tournament from each family. This generalises results of Bonnet et al. on ordered graphs. The key for these results is a polynomial time algorithm which takes as input a tournament T and computes a linear order < on V(T) such that the twin-width of the birelation (T, <) is at most some function of the twin-width of T. Since approximating twin-width can be done in FPT time for an ordered structure (T, <), this provides a FPT approximation of twin-width for tournaments.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graph enumeration
  • Tournaments
  • twin-width
  • first-order logic
  • model checking
  • NIP
  • small classes


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Nir Ailon, Moses Charikar, and Alantha Newman. Aggregating inconsistent information: Ranking and clustering. J. ACM, 55(5), November 2008. URL:
  2. Noga Alon, Graham Brightwell, H.A. Kierstead, A.V. Kostochka, and Peter Winkler. Dominating sets in k-majority tournaments. Journal of Combinatorial Theory, Series B, 96(3):374-387, 2006. URL:
  3. Mikołaj Bojańczyk. personal communication, July 2022. Google Scholar
  4. Édouard Bonnet, Dibyayan Chakraborty, Eun Jung Kim, Noleen Köhler, Raul Lopes, and Stéphan Thomassé. Twin-width VIII: delineation and win-wins. CoRR, abs/2204.00722, 2022. URL:
  5. Édouard Bonnet, Colin Geniet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width II: small classes. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA 2021), pages 1977-1996, 2021. URL:
  6. Édouard Bonnet, Ugo Giocanti, Patrice Ossona de Mendez, Pierre Simon, Stéphan Thomassé, and Szymon Toruńczyk. Twin-width IV: ordered graphs and matrices. CoRR, abs/2102.03117, 2021, to appear at STOC 2022. URL:
  7. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant. Twin-width I: tractable FO model checking. J. ACM, 69(1):3:1-3:46, 2022. URL:
  8. Édouard Bonnet, Jaroslav Nesetril, Patrice Ossona de Mendez, Sebastian Siebertz, and Stéphan Thomassé. Twin-width and permutations. CoRR, abs/2102.06880, 2021. URL:
  9. Youssef Boudabbous and Maurice Pouzet. The morphology of infinite tournaments; application to the growth of their profile. European Journal of Combinatorics, 31(2):461-481, 2010. Combinatorics and Geometry. URL:
  10. Samuel Braunfeld and Michael C Laskowski. Existential characterizations of monadic nip. arXiv preprint, 2022. URL:
  11. Maria Chudnovsky, Alexandra Fradkin, and Paul Seymour. Tournament immersion and cutwidth. J. Comb. Theory, Ser. B, 102:93-101, January 2012. URL:
  12. Rodney G Downey, Michael R Fellows, and Udayan Taylor. The parameterized complexity of relational database queries and an improved characterization of W[1]. DMTCS, 96:194-213, 1996. Google Scholar
  13. Fedor V. Fomin and Michał Pilipczuk. On width measures and topological problems on semi-complete digraphs. Journal of Combinatorial Theory, Series B, 138:78-165, 2019. URL:
  14. Alexandra Fradkin and Paul Seymour. Tournament pathwidth and topological containment. Journal of Combinatorial Theory, Series B, 103(3):374-384, 2013. URL:
  15. Jakub Gajarský, Petr Hliněný, Jan Obdržálek, Daniel Lokshtanov, and M. S. Ramanujan. A new perspective on fo model checking of dense graph classes. ACM Trans. Comput. Logic, 21(4), July 2020. URL:
  16. Colin Geniet and Stéphan Thomassé. First order logic and twin-width in tournaments and dense oriented graphs, 2022. URL:
  17. Sylvain Guillemot and Dániel Marx. Finding small patterns in permutations in linear time. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 82-101, 2014. URL:
  18. Mithilesh Kumar and Daniel Lokshtanov. Faster Exact and Parameterized Algorithm for Feedback Vertex Set in Tournaments. In Nicolas Ollinger and Heribert Vollmer, editors, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016), volume 47 of Leibniz International Proceedings in Informatics (LIPIcs), pages 49:1-49:13, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL:
  19. Adam Marcus and Gábor Tardos. Excluded permutation matrices and the Stanley-Wilf conjecture. J. Comb. Theory, Ser. A, 107(1):153-160, 2004. URL:
  20. Serguei Norine, Paul Seymour, Robin Thomas, and Paul Wollan. Proper minor-closed families are small. Journal of Combinatorial Theory, Series B, 96(5):754-757, 2006. URL:
  21. Michał Pilipczuk. Tournaments and Optimality: New Results in Parameterized Complexity. PhD thesis, University of Bergen, August 2013. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail