  Phylogenetic networks are rooted directed acyclic graphs that represent
evolutionary relationships between species whose past includes reticulation
events such as hybridisation and horizontal gene transfer. To search the space
of phylogenetic networks, the popular tree rearrangement operation rooted
subtree prune and regraft (rSPR) was recently generalised to phylogenetic
networks. This new operation - called subnet prune and regraft (SNPR) - induces
a metric on the space of all phylogenetic networks as well as on several
widely-used network classes. In this paper, we investigate several problems
that arise in the context of computing the SNPR-distance. For a phylogenetic
tree $T$ and a phylogenetic network $N$, we show how this distance can be
computed by considering the set of trees that are embedded in $N$ and then use
this result to characterise the SNPR-distance between $T$ and $N$ in terms of
agreement forests. Furthermore, we analyse properties of shortest
SNPR-sequences between two phylogenetic networks $N$ and $N'$, and answer the
question whether or not any of the classes of tree-child, reticulation-visible,
or tree-based networks isometrically embeds into the class of all phylogenetic
networks under SNPR.
