Ubiquity in graphs III: Ubiquity of locally finite graphs with extensive tree-decompositions

A graph $G$ is said to be ubiquitous, if every graph $Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every locally finite graph is ubiquitous. In this paper we show that locally finite graphs admitting a certain type of tree-decomposition, which we call an extensive tree-decomposition, are ubiquitous. In particular this includes all locally finite graphs of finite tree-width, and also all locally finite graphs with finitely many ends, all of which have finite degree. It remains an open question whether every locally finite graph admits an extensive tree-decomposition.