  The Treewidth-2 Vertex Deletion problem asks whether a set of at most $t$
vertices can be removed from a graph, such that the resulting graph has
treewidth at most two. A graph has treewidth at most two if and only if it does
not contain a $K_4$ minor. Hence, this problem corresponds to the NP-hard
$\mathcal{F}$-Minor Cover problem with $\mathcal{F} = \{K_4\}$. For any variant
of the $\mathcal{F}$-Minor Cover problem where $\mathcal{F}$ contains a planar
graph, it is known that a polynomial kernel exists. I.e., a preprocessing
routine that in polynomial time outputs an equivalent instance of size
$t^{O(1)}$. However, this proof is non-constructive, meaning that this proof
does not yield an explicit bound on the kernel size. The $\{K_4\}$-Minor Cover
problem is the simplest variant of the $\mathcal{F}$-Minor Cover problem with
an unknown kernel size.
  To develop a constructive kernelization algorithm, we present a new method to
decompose graphs into near-protrusions, such that near-protrusions in this new
decomposition can be reduced using elementary reduction rules. Our method
extends the `approximation and tidying' framework by van Bevern et al.
[Algorithmica 2012] to provide guarantees stronger than those provided by both
this framework and a regular protrusion decomposition. Furthermore, we provide
extensions of the elementary reduction rules used by the $\{K_4,
K_{2,3}\}$-Minor Cover kernelization algorithm introduced by Donkers et al.
[IPEC 2021].
  Using the new decomposition method and reduction rules, we obtain a kernel
consisting of $O(t^{41})$ vertices, which is the first constructive kernel.
This kernel is a step towards more concrete kernelization bounds for the
$\mathcal{F}$-Minor Cover problem where $\mathcal{F}$ contains a planar graph,
and our decomposition provides a potential direction to achieve these new
bounds.
