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

A structural duality for path-decompositions into parts of small radius