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 nG≼Γ for all n ∈ N, then one also has ℵ0G≼Γ, where αG 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.