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

It is an easy observation that if a graph~$G$ admits a path-decomposition whose parts have small radius, then $G$ contains no large subdivision of $K_{1,3}$ or $K^3$ as a (quasi-)geodesic subgraph. We show that these are in fact the only obstructions to such path-decompositions of small radial width, and we prove analogous results for decompositions modelled on cycles and subdivided stars instead of paths. With our results we confirm in a strong form a conjecture of Georgakopoulos and Papasoglu on fat-minor-characterisations of graphs quasi-isometric to paths, cycles and paths, and subdivided stars, respectively. For this, we present a novel view on quasi-isometries between graphs by graph-decompositions of bounded radial width and spread. This new perspective enables us to prove further results in coarse graph theory, and may thus be of independent interest.