While trying to understand Baker's approach (also explained in this article by Eppstein) to designing PTAS's for planar graphs, I came across a difficulty.
The idea is, given an integer $k$, decompose the graph into subgraphs of tree-width bounded by $O(k)$ (which for planar graphs happens if $diameter = O(k)$), then solve the problem on the subgraphs in polynomial-time, and finally glue the solutions together to get a polynomial-time $(1 + O(1/k))$-approx.
The decomposition method described in the part $8$ of the article, amounts to, as far as I understood, pick any vertex, get a BFS tree from that vertex, and remove every $k$-th level of the tree (potentially starting at level $0 \leq i < k$, but I do not think it matters for this part).
The author then argues that the obtained subgraphs (consisting each of at most $k-1$ levels of the BFS tree), have bounded diameter (i.e. depending only on $k$).
This is the part I do not understand. For instance, what happens if some layer of the tree is a long path ? How is the diameter of a subgraph containing this layer bounded ?
Thank you for any help,