A $μ$-distance for semidirected orchard phylogenetic networks

Abstract

In evolutionary biology, phylogenetic networks are now widely used to represent the historical relationships between species and population, when this history includes reticulation events such as hybridization, gene flow and admixture between populations. Semidirected phylogenetic networks are appropriate models when the direction of some edges and the root position are not identifiable from data. Comparing semidirected networks is important in many applications. For rooted and directed networks, a $μ$-representation was originally introduced to distinguish tree-child networks, and has since been extended in two different directions: to the larger class of orchard directed networks by adding an extra component that counts paths to reticulations; and to semidirected networks, through an edge-based variant. However, the latter does not provide a distance between semidirected and orchard networks. We introduce here a new edge-based $μ$-representation capable of distinguishing distinct orchard binary semidirected networks. For this class, we provide a reconstruction algorithm and therefore obtain a true distance that is computable in polynomial time.