Understanding bounded-diameter decomposition of graphs for PTAS
Actually, the obtained subgraphs may not have bounded diameter, but they still have bounded tree-width. In order to see this, take one such subgraph $H$, and build $H'$ by adding a new vertex $u$ that is connected to each of the top-level vertices of $H$ (remember that $H$ was obtained as $k-1$ consecutive layers of a BFS tree). Then clearly $diam(H') \leq 2k$ since any vertex in $H'$ is at distance less than $k$ of $u$. We then know that $tw(H') = O(diam(H')) = O(k)$. Finally, $H$ is a minor of $H'$ so $tw(H) \leq tw(H')$, hence $tw(H) = O(k)$.