Metrics for classes of semi-binary phylogenetic networks using $\mu$-representations

Phylogenetic networks are useful in representing the evolutionary history of taxa. In certain scenarios, one requires a way to compare different networks. In practice, this can be rather difficult, except within specific classes of networks. In this paper, we derive metrics for the class of \emph{orchard networks} and the class of \emph{strongly reticulation-visible} networks, from variants of so-called \emph{$\mu$-representations}, which are vector representations of networks. For both network classes, we impose degree constraints on the vertices, by considering \emph{semi-binary} networks.