Locating a Phylogenetic Tree in a Reticulation-Visible Network in Quadratic Time

In phylogenetics, phylogenetic trees are rooted binary trees, whereas phylogenetic networks are rooted arbitrary acyclic digraphs. Edges are directed away from the root and leaves are uniquely labeled with taxa in phylogenetic networks. For the purpose of validating evolutionary models, biologists check whether or not a phylogenetic tree is contained in a phylogenetic network on the same taxa. This tree containment problem is known to be NP-complete. A phylogenetic network is reticulation-visible if every reticulation node separates the root of the network from some leaves. We answer an open problem by proving that the problem is solvable in quadratic time for reticulation-visible networks. The key tool used in our answer is a powerful decomposition theorem. It also allows us to design a linear-time algorithm for the cluster containment problem for networks of this type and to prove that every galled network with n leaves has 2(n-1) reticulation nodes at most.Comment: The journal version of arXiv:1507.02119v

Counting General Phylogenetic networks

We provide precise asymptotic estimates for the number of general phylogenetic networks by using analytic combinatorial methods. Recently, this approach is studied by Fuchs, Gittenberger, and the author himself (Australasian Journal of Combinatorics 73(2):385-423, 2019), to count networks with few reticulation vertices for two subclasses: tree-child and normal networks. We follow this line of research to show how to obtain results on the enumeration of general phylogenetic networks.Comment: 36 pages, 2 Tables. arXiv admin note: text overlap with arXiv:1803.1132

Heading in the right direction? Using head moves to traverse phylogenetic network space

Head moves are a type of rearrangement moves for phylogenetic networks. They have mostly been studied as part of more encompassing types of moves, such as rSPR moves. Here, we study head moves as a type of moves on themselves. We show that the tiers ($k>0$) of phylogenetic network space are connected by local head moves. Then we show tail moves and head moves are closely related: sequences of tail moves can be converted to sequences of head moves and vice versa, changing the length by at most a constant factor. Because the tiers of network space are connected by rSPR moves, this gives a second proof of the connectivity of these tiers. Furthermore, we show that these tiers have small diameter by reproving the connectivity a third time. As the head move neighbourhood is in general small, this makes head moves a good candidate for local search heuristics. Finally we prove that finding the shortest sequence of head moves between two networks is NP-hard.Comment: 39 pages, 27 figure