Monadically dependent graph classes are those in which first-order logic cannot encode all graphs. They include all nowhere dense classes and all classes of bounded twin-width. We show that first-order model checking is fixed-parameter tractable on every monadically dependent graph class. Together with the known hardness result for hereditary classes that are not monadically dependent, this gives a complete characterization of the hereditary graph classes with fixed-parameter tractable first-order model checking. This resolves a central open problem in algorithmic model theory. Our proof combines ideas from structural graph theory (graph decompositions), finite model theory (locality of first-order logic), and model theory (counting types). In particular, we show that in monadically dependent classes, tuples of fixed length have only $|U|^{O(1)}$ types of fixed quantifier rank over any set of parameters $U$.
Monadically Dependent Graph Classes Have Almost Bounded Merge-Width
Jan Dreier, Nikolas Mählmann, Michał Pilipczuk, Szymon Toruńczyk
AI 速读 基于摘要
作者证明每个单子依赖图类都具有几乎有界合并宽度，并给出近立方时间的算法构造。
cs.DM · cs.DS · cs.LO · math.CO · math.LO
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A graph class $\mathcal{C}$ is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from $\mathcal{C}$ using a fixed formula of first-order logic. We prove that every monadically dependent graph class has almost bounded merge-width, thereby confirming a conjecture of the first and the last author [STOC '25]. Our proof is algorithmic and the witnessing decompositions in the form of merge sequences can be computed in nearly-cubic time. This is a step towards the proof of fixed-parameter tractability of model checking first-order logic on monadically dependent graph classes, which is achieved in a companion paper.