Ubiquity of graphs with nowhere-linear end structure.

A graph G is said to be ≼ - ubiquitous , where ≼ is the minor relation between graphs, if whenever Γ is a graph with n G ≼ Γ for all n ∈ N , then one also has ℵ 0 G ≼ Γ , where α G is the disjoint union of α many copies of G . A well-known conjecture of Andreae is that every locally finite connected graph is ≼ -ubiquitous. In this paper we give a sufficient condition on the structure of the ends of a graph G which implies that G is ≼ -ubiquitous. In particular this implies that the full-grid is ≼ -ubiquitous.