We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments $\mathcal T$, first-order model checking either is fixed parameter tractable, or is AW$[*]$-hard. This dichotomy coincides with the fact that $\mathcal T$ has either bounded or unbounded twin-width, and that the growth of $\mathcal T$ 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: $\mathcal T$ 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.

翻译：我们刻画了具有可处理一阶模型检验的锦标赛图类。对于每个遗传锦标赛图类 $\mathcal T$，一阶模型检验要么是固定参数可处理的，要么是 AW$[*]$-难的。这一二分性质与 $\mathcal T$ 是否具有有界或无界孪生宽度，以及 $\mathcal T$ 的增长速率至多为指数级或至少为阶乘级是一致的。从模型论角度看，我们证明锦标赛图的 NIP 类恰与有界孪生宽度相对应。孪生宽度还由三个无穷阻碍族刻画：$\mathcal T$ 具有有界孪生宽度当且仅当其排除每个族中的至少一个锦标赛图。这推广了 Bonnet 等人关于有序图的结果。这些结果的关键是一个多项式时间算法，该算法以锦标赛图 $T$ 为输入，计算 $V(T)$ 上的一个线性顺序 $<$，使得二元关系 $(T,<)$ 的孪生宽度至多为 $T$ 的孪生宽度的一个函数。由于对于有序结构 $(T,<)$，孪生宽度的近似可在多项式时间内完成，这为锦标赛图的孪生宽度提供了多项式时间近似。我们的结果可推广至具有有界稳定集的有向图，此类图还可通过任意二元关系进行扩充。