  Suppose $G$ is a graph with degrees bounded by $d$, and one needs to remove
more than $\epsilon n$ of its edges in order to make it planar. We show that in
this case the statistics of local neighborhoods around vertices of $G$ is far
from the statistics of local neighborhoods around vertices of any planar graph
$G'$ with the same degree bound. In fact, a similar result is proved for any
minor-closed property of bounded degree graphs.
  As an immediate corollary of the above result we infer that many well studied
graph properties, like being planar, outer-planar, series-parallel, bounded
genus, bounded tree-width and several others, are testable with a constant
number of queries, where the constant may depend on $\epsilon$ and $d$, but not
on the graph size. None of these properties was previously known to be testable
even with $o(n)$ queries.
