Flip-width: Cops and Robber on dense graphs

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.