Counting Phylogenetic Networks with Few Reticulation Vertices: Galled and Reticulation-Visible Networks.

We give exact and asymptotic counting results for the number of galled networks and reticulation-visible networks with few reticulation vertices. Our results are obtained with the component graph method, which was introduced by L. Zhang and his coauthors, and generating function techniques. For galled networks, we in addition use analytic combinatorics. Moreover, in an appendix, we consider maximally reticulated reticulation-visible networks and derive their number, too.

Compression of Phylogenetic Networks and Algorithm for the Tree Containment Problem.

Rooted phylogenetic networks are rooted acyclic digraphs. They are used to model complex evolution where hybridization, recombination, and other reticulation events play a role. A rigorous definition of network compression is introduced on the basis of recent studies of relationships between cluster, tree, and rooted phylogenetic networks. The concept reveals new connections between well-studied network classes, including tree-child networks and reticulation-visible networks. It also enables us to define a new class of networks for which the cluster containment problem is solvable in linear time.