On Universal Graphs With Forbidden Topological Subgraphs

If a class G of countable graphs has a member G* that contains a copy of every G ∈ G then G* is called universal in G . If every G ∈ G is isomorphic to an induced subgraph of G* we call G* strongly universal in G . By determining for which n, m ∈ ℕ the class G (TKn,m) of all countable graphs with forbidden subdivisions of Kn,m has a (strongly) universal element we prove a conjecture of R. Halin and also show that weak and strong universality are not equivalent