We characterise the classes of tournaments with tractable first-order model checking. For every hereditary class of tournaments $\mathcal T$, first-order model checking is either fixed parameter tractable or $\textrm{AW}[*]$-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 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 that 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 polynomial time for an ordered structure $(T,<)$, this provides a polynomial time 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：我们刻画了具有易处理一阶模型检测的锦标赛类。对于任意遗传锦标赛类$\mathcal T$，一阶模型检测要么是固定参数易处理的，要么是$\textrm{AW}[*]$-困难的。这一二分现象与$\mathcal T$具有有界或无界孪生宽度的事实相吻合，且$\mathcal T$的增长速率至多为指数级或至少为阶乘级。从模型论视角来看，我们证明锦标赛的NIP类恰好对应有界孪生宽度。孪生宽度还可通过三个无限障碍族来刻画：$\mathcal T$具有有界孪生宽度当且仅当它排除每个族中的至少一个锦标赛。这推广了Bonnet等人关于有序图的研究结果。这些结论的关键在于一个多项式时间算法，该算法以锦标赛$T$为输入，计算$V(T)$上的线性序$<$，使得二元关系$(T,<)$的孪生宽度至多为$T$孪生宽度的某个函数。由于对于有序结构$(T,<)$可在多项式时间内近似孪生宽度，这为锦标赛的孪生宽度提供了多项式时间近似方案。我们的结果可推广至具有有界稳定集规模的有向图，此类图还可通过任意二元关系进行扩充。