  The need for structures capable of accommodating complex evolutionary signals
such as those found in, for example, wheat has fueled research into
phylogenetic networks. Such structures generalize the standard phylogenetic
tree model by also allowing cycles and have been introduced in rooted and
unrooted form. In contrast to phylogenetic trees, however, surprisingly little
is known about the interplay between both types thus hampering our ability to
make much needed progress for rooted phylogenetic networks by drawing on
insights from their much better understood unrooted counterparts. Unrooted
phylogenetic networks are underpinned by split systems and by focusing on them
we establish a first link between both types. More precisely, we develop a link
between 1-nested phylogenetic networks which are examples of rooted
phylogenetic networks and the well-studied median networks (aka Buneman graph)
which are examples of unrooted phylogenetic networks. In particular, we show
that not only can a 1-nested network be obtained from a median network but also
that that network is, in a well-defined sense, optimal. Along the way, we
characterize circular split systems in terms of the novel $\mathcal
I$-intersection closure of a split system and establish the 1-nested analogue
of the fundamental "Splits Equivalence Theorem" for phylogenetic trees.
