This paper studies the relationship between undirected (unrooted) and
directed (rooted) phylogenetic networks. We describe a polynomial-time
algorithm for deciding whether an undirected binary phylogenetic network, given
the locations of the root and reticulation vertices, can be oriented as a
directed phylogenetic network. Moreover, we give a mathematical
characterization of when this is the case and show that this directed
phylogenetic network is then always unique. These results are generalized to
the nonbinary case. In addition, we describe an algorithm for deciding whether
an undirected binary phylogenetic network can be oriented as a directed
phylogenetic network of a certain class. The algorithm is fixed-parameter
tractable (FPT) when the parameter is the level of the network and is
applicable to a wide range of network classes, including tree-child,
tree-based, stack-free and orchard networks. It can also be used to decide
whether an undirected phylogenetic network is tree-based and whether a
partly-directed phylogenetic network can be oriented as a directed phylogenetic
network. Finally, we show that, in general, it is NP-hard to decide whether an
undirected network can be oriented as a tree-based network