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arXiv · 2611.06804

Monadically Dependent Graph Classes Have Almost Bounded Merge-Width

Jan Dreier·Nikolas Mählmann·Michał Pilipczuk·Szymon Toruńczyk
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Abstract

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.

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Jan Dreier, Nikolas Mählmann, Michał Pilipczuk, Szymon Toruńczyk. 2026-11-04. Monadically Dependent Graph Classes Have Almost Bounded Merge-Width. https://arxiv.org/abs/2611.06804

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