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Suppose $G$ is a graph with degrees bounded by $d$, and one needs to removemore than $\epsilon n$ of its edges in order to make it planar. We show that inthis case the statistics of local neighborhoods around vertices of $G$ is farfrom 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 anyminor-closed property of bounded degree graphs.  As an immediate corollary of the above result we infer that many well studiedgraph properties, like being planar, outer-planar, series-parallel, boundedgenus, bounded tree-width and several others, are testable with a constantnumber of queries, where the constant may depend on $\epsilon$ and $d$, but noton the graph size. None of these properties was previously known to be testableeven with $o(n)$ queries.

Every Minor-Closed Property of Sparse Graphs is Testable