  Phylogenetic networks are used to represent the evolutionary history of
species. They are versatile when compared to traditional phylogenetic trees, as
they capture more complex evolutionary events such as hybridization and
horizontal gene transfer. Distance-based methods such as the Neighbor-Net
algorithm are widely used to compute phylogenetic networks from data. However,
the output is necessarily an undirected graph, posing a great challenge to
deduce the direction of genetic flow in order to infer the true evolutionary
history. Recently, Huber et al. investigated two different computational
problems relevant to orienting undirected phylogenetic networks into directed
ones. In this paper, we consider the problem of orienting an undirected binary
network into a tree-child network. We give some necessary conditions for
determining the tree-child orientability, such as a tight upper bound on the
size of tree-child orientable graphs, as well as many interesting examples. In
addition, we introduce new families of undirected phylogenetic networks, the
jellyfish graphs and ladder graphs, that are orientable but not tree-child
orientable. We also prove that any ladder graph can be made tree-child
orientable by adding extra leaves, and describe a simple algorithm for
orienting a ladder graph to a tree-child network with the minimum number of
extra leaves. We pose many open problems as well.
