  Semi-directed phylogenetic networks have recently emerged as a class of
phylogenetic networks sitting between rooted (directed) and unrooted
(undirected) phylogenetic networks as they contain both directed as well as
undirected edges. While the spaces of rooted phylogenetic networks and unrooted
phylogenetic networks have been analyzed in recent years and various
rearrangement moves to traverse these spaces have been introduced, the results
do not immediately carry over to semi-directed phylogenetic networks. Here, we
propose a simple rearrangement move for semi-directed phylogenetic networks,
called cut edge transfer (CET), and show that the space of semi-directed
level-$1$ networks with precisely $k$ reticulations is connected under CET.
This level-$1$ space is currently the predominantly used search space for most
algorithms that reconstruct semi-directed phylogenetic networks. Hence, every
semi-directed level-$1$ network with a fixed number of reticulations and leaf
set can be reached from any other such network by a sequence of CETs. By
introducing two additional moves, CET$^+$ and CET$^-$, that allow for the
addition or deletion of reticulations, we then establish connectedness for the
space of all semi-directed phylogenetic networks on a fixed leaf set. As a
byproduct of our results for semi-directed phylogenetic networks, we also show
that the space of rooted level-$1$ networks with a fixed number of
reticulations and leaf set is connected under CET, when translated into the
rooted setting.
