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Proper Minor-Closed Classes of Graphs have Assouad-Nagata Dimension 2
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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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An improved quasi-isometry between graphs of bounded cliquewidth and graphs of bounded treewidth
Every graph of cliquewidth k admits a dominated partition whose quotient has treewidth k-1, yielding a 3-quasi-isometry and Assouad-Nagata dimension 1.
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