This research shows that proper minor-closed classes exhibit Assouad–Nagata dimension 2, highlighting implications for graph structure and treewidth.
Asymptotic dimension and Assouad–Nagata dimension are measures of the large-scale shape of a class of graphs. Bonamy, Bousquet, Esperet, Groenland, Liu, Pirot, and Scott [ J. Eur. Math. Society ] showed that any proper minor-closed class has asymptotic dimension 2, dropping to 1 only if the treewidth is bounded. We improve this result by showing it also holds for the stricter Assouad–Nagata dimension. We also characterise when subdivision-closed classes of graphs have bounded Assouad–Nagata dimension.
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Marc Distel (2025) studied this question.
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