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.