  Phylogenetic networks are a generalization of phylogenetic trees that are
used to represent reticulate evolution. Unrooted phylogenetic networks form a
special class of such networks, which naturally generalize unrooted
phylogenetic trees. In this paper we define two operations on unrooted
phylogenetic networks, one of which is a generalization of the well-known
nearest-neighbor interchange (NNI) operation on phylogenetic trees. We show
that any unrooted phylogenetic network can be transformed into any other such
network using only these operations. This generalizes the well-known fact that
any phylogenetic tree can be transformed into any other such tree using only
NNI operations. It also allows us to define a generalization of tree space and
to define some new metrics on unrooted phylogenetic networks. To prove our main
results, we employ some fascinating new connections between phylogenetic
networks and cubic graphs that we have recently discovered. Our results should
be useful in developing new strategies to search for optimal phylogenetic
networks, a topic that has recently generated some interest in the literature,
as well as for providing new ways to compare networks.
