  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.
