We define new graph parameters, called flip-width, that generalize treewidth, degeneracy, and generalized coloring numbers for sparse graphs, and clique-width and twin-width for dense graphs. The flip-width 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 then 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}etril and Ossona de Mendez, and includes classes of bounded twin-width of Bonnet, Kim, Thomass{\'e}, and Watrigant. This unifies Sparsity Theory and Twin-width Theory, providing a common language for studying the central notions of the two theories, such as weak coloring numbers and twin-width -- corresponding to winning strategies of one player -- or dense shallow minors, rich divisions, or well-linked sets, corresponding to winning strategies of the other player. 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. Finally, we propose a more general notion of tameness, called almost bounded flip-width, which is a dense counterpart of nowhere dense classes. We conjecture, and provide evidence, that classes with almost bounded flip-width coincide with monadically dependent (or monadically NIP) classes, introduced by Shelah in model theory. We also provide evidence that classes of almost bounded flip-width characterise the hereditary graph classes for which the model-checking problem is fixed-parameter tractable.

翻译：我们定义了新的图参数——翻转宽度（flip-width），它统一推广了稀疏图的树宽、退化度和广义染色数，以及稠密图的团宽（clique-width）和孪生宽（twin-width）。翻转宽度参数基于警察与强盗博弈的变体定义，其中强盗的速度受固定常数 $r\in\mathbb N\cup\{\infty\}$ 限制，而警察则对目标图执行翻转（或扰动）操作。我们进一步提出图类有界性的新概念——有界翻转宽度，这是Nešetřil和Ossona de Mendez提出的有界扩张图类在稠密图上的对应，并包含Bonnet、Kim、Thomassé和Watrigant提出的有界孪生宽度图类。该理论统一了稀疏性理论与孪生宽度理论，为研究两者的核心概念提供了共同语言，例如弱染色数与孪生宽度（对应一方玩家的获胜策略），以及稠密浅层子式、丰富划分或强连通集（对应另一方玩家的获胜策略）。我们证明了有界翻转宽度在一阶解释（或转导）下保持封闭，推广了先前关于有界扩张图类与有界孪生宽度图类的结果。我们还提出了一个近似计算给定图翻转宽度的算法，其运行时间在图的规模上属于切片多项式时间（XP）。最后，我们提出了更广义的有界性概念——几乎有界翻转宽度，这是无稠密图类在稠密图上的对应。我们猜想（并提供证据表明）具有几乎有界翻转宽度的图类与Shelah在模型论中引入的单调依赖（或单调NIP）图类等价。同时提供证据表明，几乎有界翻转宽度图类刻画了模型检测问题具有固定参数可解性的遗传图类。