← 研踪 PaperTrailarXiv:2611.06804 · v1
Monadically Dependent Graph Classes Have Almost Bounded Merge-Width
首次提交 2026-11-05 01:20
版本更新 2026-11-05 01:20
雷达观测 2026-11-05 12:12
摘要
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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v1 · 2026-11-05 01:20 · 最新观测版本
Monadically Dependent Graph Classes Have Almost Bounded Merge-Width
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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