Proper Minor-Closed Classes of Graphs have Assouad-Nagata Dimension 2
Marc Distel
TL;DR
This paper proves that proper minor-closed classes of graphs have Assouad-Nagata dimension 2, extending previous results on asymptotic dimension, and characterizes when subdivision-closed classes have bounded Assouad-Nagata dimension.
Contribution
It establishes that proper minor-closed classes of graphs have Assouad-Nagata dimension 2, and characterizes conditions for boundedness in subdivision-closed classes.
Findings
Proper minor-closed classes have Assouad-Nagata dimension 2.
Bounded Assouad-Nagata dimension occurs under specific subdivision-closure conditions.
The result extends previous asymptotic dimension findings to Assouad-Nagata dimension.
Abstract
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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Taxonomy
TopicsGraph theory and applications · Advanced Graph Theory Research · Stochastic processes and statistical mechanics