  Assouad-Nagata dimension addresses both large and small scale behaviors of
metric spaces and is a refinement of Gromov's asymptotic dimension. A metric
space $M$ is a minor-closed metric if there exists an (edge-)weighted graph $G$
satisfying a fixed minor-closed property such that the underlying space of $M$
is the vertex-set of $G$, and the metric of $M$ is the distance function in
$G$. Minor-closed metrics naturally arise when removing redundant edges of the
underlying graphs by using edge-deletion and edge-contraction. In this paper,
we determine the Assouad-Nagata dimension of every minor-closed metric. It is a
common generalization of known results about the asymptotic dimension of
$H$-minor free unweighted graphs and about the Assouad-Nagata dimension of
complete Riemannian surfaces with finite Euler genus and their corollaries.

Combinatorics;Discrete Mathematics;Group Theory;Geometric Topology;Metric Geometry