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We show that many graphs with bounded treewidth can be described as subgraphsof the strong product of a graph with smaller treewidth and a bounded-sizecomplete 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 somefunction $f$, 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 graphsand show they determine the underlying treewidth of any graph class. Using thisresult, 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 underlyingtreewidth ${t-2}$. In general, we prove that a monotone class has boundedunderlying treewidth if and only if it excludes some fixed topological minor.We also study the underlying treewidth of graph classes defined by an excludedsubgraph 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.

Product structure of graph classes with bounded treewidth