Asymptotic dimension of graphs excluding an induced or fat minor

Asymptotic dimension is a large-scale invariant of metric spaces that was introduced by Gromov (1993). A major open problem in coarse graph theory asks whether, for every fixed graph $H$, the class of graphs excluding $H$ as a fat minor has bounded asymptotic dimension. A weaker conjecture asserts the same conclusion for graphs excluding $H$ as an induced minor. We substantially expand the range of graphs $H$ for which these conjectures are known to hold. Specifically, we prove bounded asymptotic dimension for the following classes of graphs: graphs excluding a fixed planar graph as a fat minor; and graphs excluding a fixed string graph as an induced minor. All of our results also hold for the stronger notion of Assouad–Nagata dimension, a refinement of asymptotic dimension. These results have a number of applications, two of which we mention here. First they imply that an induced minor-closed class is $\chi$-bounded if and only if it does not contain some Burling graph. Second they imply that, in graph classes excluding a fixed planar graph as an induced minor, treewidth is polynomial in maximum degree.