  Given an arbitrary class $\mathcal{H}$ of graphs, we investigate which graphs
admit a decomposition modelled on a graph in $\mathcal{H}$ into parts of small
radius. The $\mathcal{H}$-decompositions that we consider here generalise the
notion of tree-decompositions. We identify obstructions to such
$\mathcal{H}$-decompositions of small radial width, and we prove that these
obstructions occur in every graph of sufficiently large radial
$\mathcal{H}$-width for the classes $\mathcal{H}$ of paths, of cycles and of
subdivided stars.
