  We study treewidth sparsifiers. Informally, given a graph $G$ of treewidth
$k$, a treewidth sparsifier $H$ is a minor of $G$, whose treewidth is close to
$k$, $|V(H)|$ is small, and the maximum vertex degree in $H$ is bounded.
Treewidth sparsifiers of degree $3$ are of particular interest, as routing on
node-disjoint paths, and computing minors seems easier in sub-cubic graphs than
in general graphs.
  In this paper we describe an algorithm that, given a graph $G$ of treewidth
$k$, computes a topological minor $H$ of $G$ such that (i) the treewidth of $H$
is $\Omega(k/\text{polylog}(k))$; (ii) $|V(H)| = O(k^4)$; and (iii) the maximum
vertex degree in $H$ is $3$. The running time of the algorithm is polynomial in
$|V(G)|$ and $k$. Our result is in contrast to the known fact that unless $NP
\subseteq coNP/{\sf poly}$, treewidth does not admit polynomial-size kernels.
One of our key technical tools, which is of independent interest, is a
construction of a small minor that preserves node-disjoint routability between
two pairs of vertex subsets. This is closely related to the open question of
computing small good-quality vertex-cut sparsifiers that are also minors of the
original graph.
