Ubiquity in graphs II: 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 $\mathrm{\Gamma }$ is a graph with $nG \preceq \Gamma$forall$n \in $\mathbb{N}$$,thenonealsohas$\aleph_0 G \preceq \Gamma$,where$\alpha G$isthedisjointunionof$\alpha$manycopiesof$G$. 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.