Overlapping BFS Layers in Dominating Set with Baker's Technique
In a single layer of the partition, consisting of the vertices at distance $d$ to $d+k$ from the root, the vertices at distance $d+1$ through $d+k-1$ can be dominated the same way as they are in the whole graph, but you have no control of the size of the dominating sets of the vertices at distances $d$ or $d+k$, on the boundary of the layer: in the original graph, those vertices could be dominated by a small number of vertices in a different layer that aren't there when you restrict to that layer. So the solution is to find a system of layers with the property that the subsets of vertices at distance $d+1$ through $d+k-1$ within each layer form a partition of the whole graph, and then within each layer to only try to dominate that subset. To do this, you need to overlap the layers by two levels.