Search

Let W be any wheel graph and G the class of all countable graphs not containing W as a minor. We show that there exists a graph in G which contains every graph in G as an induced subgraph. [ABSTRACT FROM AUTHOR]

A subgraph‐universal graph/a topological minor‐universal graph in a class of graphs G ${\mathscr{G}}$ is a graph in G ${\mathscr{G}}$ which contains every graph in G ${\mathscr{G}}$ as a subgraph/topological minor. We prove that the class P ${\mathscr{P}}$ 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 P ${\mathscr{P}}$. Furthermore, we characterise for which subdivided stars T $T$ there is a topological minor‐universal graph in the class of all countable T $T$‐free graphs. [ABSTRACT FROM AUTHOR]