🤖 AI Summary
This paper establishes a structural theory for monadically stable graph classes, addressing the absence of a dense-graph analogue to the “nowhere dense” characterization. To this end, the authors introduce the Flipper–Connector game: Flipper may flip all edges between any two vertex subsets, while Connector is restricted to interactions within bounded-radius balls; monadic stability is precisely characterized by Flipper having a winning strategy in this game. Key contributions include: (i) the first game-theoretic characterization of monadic stability; (ii) an elucidation of its fundamental connection to first-order type definability; (iii) a proof that monadic stability is equivalent to existential monadic stability; (iv) a polynomial-time algorithm computing Flipper’s winning strategy; and (v) two independent proofs—one model-theoretic and one combinatorial—yielding a novel derivation of the Braunfeld–Laskowski theorem.
📝 Abstract
A class of graphs $mathscr{C}$ is monadically stable if for any unary expansion $widehat{mathscr{C}}$ of $mathscr{C}$, one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from $widehat{mathscr{C}}$. It is known that nowhere dense graph classes are monadically stable; these encompass most of the studied concepts of sparsity in graphs, including graph classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs. For several years, it has been suspected that one can create a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting. In this work we provide a step in this direction by giving a characterization of monadic stability through the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round localizes the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J.ACM'17). We give two different proofs of our main result. The first proof uses tools from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we give an alternative proof of the recent result of Braunfeld and Laskowski (arXiv 2209.05120) that monadic stability for graph classes coincides with existential monadic stability. The second proof relies on the recently introduced notion of flip-wideness (Dreier, M""ahlmann, Siebertz, and Toru'nczyk, ICALP 2023) and provides an efficient algorithm to compute Flipper's moves in a winning strategy.
