We define new graph parameters that generalize tree-width, degeneracy, and generalized coloring numbers for sparse graphs, and clique-width and twin-width for dense graphs. Those parameters are defined using variants of the Cops and Robber game, in which the robber has speed bounded by a fixed constant $r\in\mathbb N\cup\{\infty\}$, and the cops perform flips (or perturbations) of the considered graph. We propose a new notion of tameness of a graph class, called bounded flip-width, which is a dense counterpart of classes of bounded expansion of Ne\v{s}et\v{r}il and Ossona de Mendez, and includes classes of bounded twin-width of Bonnet, Kim, Thomass\'e and Watrigant. We prove that boundedness of flip-width is preserved by first-order interpretations, or transductions, generalizing previous results concerning classes of bounded expansion and bounded twin-width. We provide an algorithm approximating the flip-width of a given graph, which runs in slicewise polynomial time (XP) in the size of the graph. We also propose a more general notion of tameness, called almost bounded flip-width, which is a dense counterpart of nowhere dense classes, and includes all structurally nowhere dense classes. We conjecture, and provide evidence, that classes with almost bounded flip-width coincide with monadically dependent classes, introduced by Shelah in model theory.

翻译：摘要：我们定义了新的图参数，这些参数概括了稀疏图的树宽、退化度和广义染色数，以及稠密图的团宽与孪生宽。这些参数通过警察与强盗游戏的变体定义，其中强盗的速度由固定常数 $r\in\mathbb N\cup\{\infty\}$ 限制，而警察对目标图执行翻转（或扰动）操作。我们提出了一种新的图类驯服性概念，称为有界翻转宽度，这是 Nešetřil 和 Ossona de Mendez 提出的有界扩张类的稠密对应，并包含 Bonnet、Kim、Thomassé 和 Watrigant 提出的有界孪生宽类。我们证明翻转宽度的有界性在一阶解释（或转导）下保持不变，这推广了关于有界扩张类与有界孪生宽类的既有结论。我们提出了一种近似给定图翻转宽度的算法，该算法在图的规模上运行于切片多项式时间（XP）内。此外，我们提出了更一般的驯服性概念——几乎有界翻转宽度，这是无处稠密类的稠密对应，并包含所有结构无处稠密类。我们推测（并提供证据）几乎有界翻转宽度类与 Shelah 在模型论中引入的单演依赖类一致。