  We show that many graphs with bounded treewidth can be described as subgraphs
of the strong product of a graph with smaller treewidth and a bounded-size
complete graph. To this end, define the "underlying treewidth" of a graph class
$\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some
function $f$, for every graph ${G \in \mathcal{G}}$ there is a graph $H$ with
${\text{tw}(H) \leq c}$ such that $G$ is isomorphic to a subgraph of ${H
\boxtimes K_{f(\text{tw}(G))}}$. We introduce disjointed coverings of graphs
and show they determine the underlying treewidth of any graph class. Using this
result, we prove that the class of planar graphs has underlying treewidth 3;
the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for ${t
\geq \max\{s,3\}}$); and the class of $K_t$-minor-free graphs has underlying
treewidth ${t-2}$. In general, we prove that a monotone class has bounded
underlying treewidth if and only if it excludes some fixed topological minor.
We also study the underlying treewidth of graph classes defined by an excluded
subgraph or excluded induced subgraph. We show that the class of graphs with no
$H$ subgraph has bounded underlying treewidth if and only if every component of
$H$ is a subdivided star, and that the class of graphs with no induced $H$
subgraph has bounded underlying treewidth if and only if every component of $H$
is a star.
