  Treewidth is a graph parameter that plays a fundamental role in several
structural and algorithmic results. We study the problem of decomposing a given
graph $G$ into node-disjoint subgraphs, where each subgraph has sufficiently
large treewidth. We prove two theorems on the tradeoff between the number of
the desired subgraphs $h$, and the desired lower bound $r$ on the treewidth of
each subgraph. The theorems assert that, given a graph $G$ with treewidth $k$,
a decomposition with parameters $h,r$ is feasible whenever $hr^2 \le
k/\polylog(k)$, or $h^3r \le k/\polylog(k)$ holds. We then show a framework for
using these theorems to bypass the well-known Grid-Minor Theorem of Robertson
and Seymour in some applications. In particular, this leads to substantially
improved parameters in some Erdos-Posa-type results, and faster algorithms for
a class of fixed-parameter tractable problems.
