Huynh et al. recently showed that a countable graph G which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds if G contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and Kühn. Moreover, we construct a locally finite planar graph which contains every locally finite planar graph as a topological minor. This shows that in the above result it is not enough to require that G contains every locally finite planar graph as a topological minor.

黄等人。最近表明，包含每个可数平面图作为子图的可数图G必须包含任意大的有限完全图作为拓扑子图，以及无限完全图作为子图。我们通过证明如果G包含每个可数平面图作为拓扑小数，则同样的结论成立，从而强化了这一结果。特别是，不存在包含每个可数平面图作为拓扑次要的可数平面图，回答了迪斯特尔和库恩的问题。此外，我们构造了一个局部有限平面图，其中包含每个局部有限平面图作为拓扑次要图。这表明，在上述结果中，要求G包含每个局部有限平面图作为拓扑小数是不够的。