  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.
