Phylogenetic network classes through the lens of expanding covers

It was recently shown that a large class of phylogenetic networks, the 'labellable' networks, is in bijection with the set of 'expanding' covers of finite sets. In this paper, we show how several prominent classes of phylogenetic networks can be characterised purely in terms of properties of their associated covers. These classes include the tree-based, tree-child, orchard, tree-sibling, and normal networks. In the opposite direction, we give an example of how a restriction on the set of expanding covers can define a new class of networks, which we call 'spinal' phylogenetic networks.