Universal graphs for the topological minor relation

TL;DR

This paper proves that no single countable planar graph can contain all other countable planar graphs as topological minors, and characterizes when topological minor-universal graphs exist in T-free graph classes.

Contribution

It resolves an open question by showing the non-existence of topological minor-universal graphs in countable planar graphs and characterizes such universality for T-free graphs.

Findings

No topological minor-universal graph exists for all countable planar graphs.

Characterization of subdivided stars T for which T-free graphs have a topological minor-universal graph.

Strengthens previous results on subgraph-universality in planar graphs.

Abstract

A subgraph-universal graph/a topological minor-universal graph in a class of graphs $\mathcal{G}$ is a graph in $\mathcal{G}$ which contains every graph in $\mathcal{G}$ as a subgraph/topological minor. We prove that the class $\mathcal{P}$ of all countable planar graphs does not contain a topological minor-universal graph. This answers a question of Diestel and K\"uhn and strengthens a result of Pach stating that there is no subgraph-universal graph in $\mathcal{P}$. Furthermore, we characterise for which subdivided stars $T$ there is a topological minor-universal graph in the class of all countable $T$-free graphs.