  Popular methods for exploring the space of rooted phylogenetic trees use
rearrangement moves such as rNNI (rooted Nearest Neighbour Interchange) and
rSPR (rooted Subtree Prune and Regraft). Recently, these moves were generalized
to rooted phylogenetic networks, which are a more suitable representation of
reticulate evolutionary histories, and it was shown that any two rooted
phylogenetic networks of the same complexity are connected by a sequence of
either rSPR or rNNI moves. Here, we show that this is possible using only tail
moves, which are a restricted version of rSPR moves on networks that are more
closely related to rSPR moves on trees. The connectedness still holds even when
we restrict to distance-1 tail moves (a localized version of tail-moves).
Moreover, we give bounds on the number of (distance-1) tail moves necessary to
turn one network into another, which in turn yield new bounds for rSPR, rNNI
and SPR (i.e. the equivalent of rSPR on unrooted networks). The upper bounds
are constructive, meaning that we can actually find a sequence with at most
this length for any pair of networks. Finally, we show that finding a shortest
sequence of tail or rSPR moves is NP-hard.
