Flipper games for monadically stable graph classes
Jakub Gajarsk\'y, Nikolas M\"ahlmann, Rose McCarty, Pierre Ohlmann,, Micha{\l} Pilipczuk, Wojciech Przybyszewski, Sebastian Siebertz, Marek, Soko{\l}owski, Szymon Toru\'nczyk

TL;DR
This paper characterizes monadically stable graph classes using a Flipper game, connecting model-theoretic properties with a combinatorial game, and provides two proofs including an algorithmic approach.
Contribution
It introduces a Flipper game characterization of monadic stability in graphs, bridging model theory and combinatorial game theory, with two distinct proofs and an algorithmic component.
Findings
Characterization of monadic stability via the Flipper game
Equivalence of monadic stability and a game-theoretic property
An efficient algorithm for Flipper's moves in the game
Abstract
A class of graphs is monadically stable if for any unary expansion of , one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from . 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…
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