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It was recently shown that a large class of phylogenetic networks, the`labellable' networks, is in bijection with the set of `expanding' covers offinite sets. In this paper, we show how several prominent classes ofphylogenetic networks can be characterised purely in terms of properties oftheir associated covers. These classes include the tree-based, tree-child,orchard, tree-sibling, and normal networks.

Phylogenetic network classes through the lens of expanding covers