Universal graphs for the topological minor relation

A subgraph-universal graph/a topological minor-universal graph in a class of graphs (Figure presented.) is a graph in (Figure presented.) which contains every graph in (Figure presented.) as a subgraph/topological minor. We prove that the class (Figure presented.) of all countable planar graphs does not contain a topological minor-universal graph. This answers a question of Diestel and Kühn and strengthens a result of Pach stating that there is no subgraph-universal graph in (Figure presented.). Furthermore, we characterise for which subdivided stars (Figure presented.) there is a topological minor-universal graph in the class of all countable (Figure presented.) -free graphs.