  Let $G$ be an undirected, bounded degree graph with $n$ vertices. Fix a
finite graph $H$, and suppose one must remove $\varepsilon n$ edges from $G$ to
make it $H$-minor free (for some small constant $\varepsilon > 0$). We give an
$n^{1/2+o(1)}$-time randomized procedure that, with high probability, finds an
$H$-minor in such a graph. As an application, suppose one must remove
$\varepsilon n$ edges from a bounded degree graph $G$ to make it planar. This
result implies an algorithm, with the same running time, that produces a
$K_{3,3}$ or $K_5$ minor in $G$. No prior sublinear time bound was known for
this problem.
  By the graph minor theorem, we get an analogous result for any minor-closed
property. Up to $n^{o(1)}$ factors, this resolves a conjecture of
Benjamini-Schramm-Shapira (STOC 2008) on the existence of one-sided property
testers for minor-closed properties. Furthermore, our algorithm is nearly
optimal, by an $\Omega(\sqrt{n})$ lower bound of Czumaj et al (RSA 2014).
  Prior to this work, the only graphs $H$ for which non-trivial one-sided
property testers were known for $H$-minor freeness are the following: $H$ being
a forest or a cycle (Czumaj et al, RSA 2014), $K_{2,k}$, $(k\times 2)$-grid,
and the $k$-circus (Fichtenberger et al, Arxiv 2017).
