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First Order Logic and Twin-Width in Tournaments

Authors Colin Geniet , Stéphan Thomassé

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  • 14 pages

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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.

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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


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