Ubiquity of graphs with nowhere-linear end structure

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