CSD homomorphisms between phylogenetic networks

Since Darwin, species trees have been used as a simplified description of the relationships which summarize the complicated network N of reality. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are situations where trees are inadequate. Consequently it is important to determine properties that characterize networks closely related to N and possibly more complicated than trees but lacking the full complexity of N. A connected surjective digraph map (CSD) is a map f from one network N to another network M such that every arc is either collapsed to a single vertex or is taken to an arc, such that f is surjective, and such that the inverse image of a vertex is always connected. CSD maps are shown to behave well under composition. It is proved that if there is a CSD map from N to M, then there is a way to lift an undirected version of M into N, often with added resolution. A CSD map from N to M puts strong constraints on N. In general, it may be useful to study classes of networks such that, for any N, there exists a CSD map from N to some standard member of that class.