Flip-width: Cops and Robber on dense graphs
Szymon Toruńczyk
TL;DR
This work introduces flip-width, a dense-graph analogue of degeneracy, treewidth, and generalized coloring numbers, defined via flipper games that use controlled edge flips. It unifies Sparsity Theory and Twin-width Theory by showing that bounded flip-width captures both bounded expansion and bounded twin-width, and that fw$_ infty$ aligns with clique-width, while fw$_1$ aligns with degeneracy in sparse contexts. The authors prove that bounded flip-width is preserved under first-order interpretations and FO transductions, and they provide an XP-approximation algorithm for fw, enabling practical engagement with the parameter. They further propose almost bounded flip-width as a dense surrogate for nowhere dense classes, conjecturing equivalence with monadically dependent classes and connecting to fixed-parameter tractable model checking; they also explore ordered graphs, twin-width equivalences, and several closure properties. Together, these results establish flip-width as a foundational framework for dense graph tameness, with broad implications for algorithmic meta-theorems, transduction-closed classes, and model-checking tractability across sparse and dense regimes.
Abstract
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 Nešetril and Ossona de Mendez, and includes classes of bounded twin-width of Bonnet, Kim, Thomass{é}, 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.