First Order Logic and Twin-Width in Tournaments and Dense Oriented Graphs
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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 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 compute 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 polynomial time for an ordered structure (T,<), this provides a polytime approximation of twin-width for tournaments. Our results extend to oriented graphs with stable sets of bounded size, which may also be augmented by arbitrary binary relations.
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