  In the $\mathcal{F}$-Minor-Free Deletion problem one is given an undirected
graph $G$, an integer $k$, and the task is to determine whether there exists a
vertex set $S$ of size at most $k$, so that $G-S$ contains no graph from the
finite family $\mathcal{F}$ as a minor. It is known that whenever $\mathcal{F}$
contains at least one planar graph, then $\mathcal{F}$-Minor-Free Deletion
admits a polynomial kernel, that is, there is a polynomial-time algorithm that
outputs an equivalent instance of size $k^{\mathcal{O}(1)}$ [Fomin, Lokshtanov,
Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive
arguments based on well-quasi-ordering and does not provide a concrete bound on
the kernel size.
  We study the Outerplanar Deletion problem, in which we want to remove at most
$k$ vertices from a graph to make it outerplanar. This is a special case of
$\mathcal{F}$-Minor-Free Deletion for the family $\mathcal{F} = \{K_4,
K_{2,3}\}$. The class of outerplanar graphs is arguably the simplest class of
graphs for which no explicit kernelization size bounds are known. By exploiting
the combinatorial properties of outerplanar graphs we present elementary
reduction rules decreasing the size of a graph. This yields a constructive
kernel with $\mathcal{O}(k^4)$ vertices and edges. As a corollary, we derive
that any minor-minimal obstruction to having an outerplanar deletion set of
size $k$ has $\mathcal{O}(k^4)$ vertices and edges.
