A class of graphs $\mathscr{C}$ is monadically stable if for any unary expansion $\widehat{\mathscr{C}}$ of $\mathscr{C}$, one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from $\widehat{\mathscr{C}}$. It is known that nowhere dense graph classes are monadically stable; these encompass most of the studied concepts of sparsity in graphs, including graph classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs. For several years, it has been suspected that one can create a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting. In this work we provide a step in this direction by giving a characterization of monadic stability through the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round localizes the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J.ACM'17). We give two different proofs of our main result. The first proof uses tools from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we give an alternative proof of the recent result of Braunfeld and Laskowski (arXiv 2209.05120) that monadic stability for graph classes coincides with existential monadic stability. The second proof relies on the recently introduced notion of flip-wideness (Dreier, M\"ahlmann, Siebertz, and Toru\'nczyk, arXiv 2206.13765) and provides an efficient algorithm to compute Flipper's moves in a winning strategy.

翻译：图类 $\mathscr{C}$ 是一元稳定的，如果对于 $\mathscr{C}$ 的任意一元扩张 $\widehat{\mathscr{C}}$，无法用一阶逻辑在 $\widehat{\mathscr{C}}$ 的图中解释任意长的线性序。已知无处稀见图类是一元稳定的；这类图涵盖了图中稀疏性的绝大多数研究概念，包括排除固定拓扑子式的图类。另一方面，一元稳定性是纯粹以模型论术语表述的性质，因此也适用于捕捉稠密图中的结构。多年来，学界猜测可以为一元稳定图类建立一种结构理论，在稠密背景下镜像无处稀见图类的理论。在本工作中，我们通过翻转游戏对一元稳定性进行刻画，朝着这一方向迈出了一步：该游戏由翻转者（Flipper）和连接者（Connector）在图上进行，翻转者每轮可任意互补两个顶点子集之间的边关系，连接者每轮将游戏限定在有限半径的球内。这类似于刻画无处稀见图类的分裂者游戏（Grohe, Kreutzer, and Siebertz, J.ACM'17）。我们给出了主要结果的两种不同证明。第一种证明使用模型论工具，揭示了一元稳定图类一个与类型可定义性精神相近的额外性质。作为副产品，我们给出了Braunfeld和Laskowski（arXiv 2209.05120）近期结果的另一种证明：图类的一元稳定性等价于存在一元稳定性。第二种证明依赖于最近引入的翻转宽性概念（Dreier, Mählmann, Siebertz, and Toruńczyk, arXiv 2206.13765），并提供了计算翻转者在必胜策略中步子的高效算法。