On encodings of phylogenetic networks of bounded level

Abstract

Phylogenetic networks have now joined phylogenetic trees in the center of phylogenetics research. Like phylogenetic trees, such networks canonically induce collections of phylogenetic trees, clusters, and triplets, respectively. Thus it is not surprising that many network approaches aim to reconstruct a phylogenetic network from such collections. Related to the well-studied perfect phylogeny problem, the following question is of fundamental importance in this context: When does one of the above collections encode (i.e. uniquely describe) the network that induces it? For the large class of level-1 (phylogenetic) networks we characterize those level-1 networks for which an encoding in terms of one (or equivalently all) of the above collections exists. In addition, we show that three known distance measures for comparing phylogenetic networks are in fact metrics on the resulting subclass and give the diameter for two of them. Finally, we investigate the related concept of indistinguishability and also show that many properties enjoyed by level-1 networks are not satisfied by networks of higher level.

Trinets encode tree-child and level-2 phylogenetic networks

“…However, these algorithms share a common weakness in that, even if all of the triplets within a given network are taken as input, there is no guarantee that the original network will be reconstructed. This is because, in contrast to trees, the triplets in a network do not necessarily encode the network [13]. For example, Figure 1 presents three different networks that all contain the same set of triplets.…”

“…Such a network is called binary if all vertices have indegree and outdegree at most two and all vertices with indegree two have outdegree one. In addition, a binary network is called level -k [11,12,13,14,18,19] if each biconnected component has at most k indegree-2 vertices, and it is called tree-child [6,8,21,29] if each non-leaf vertex has at least one child which has indegree 1. Note that a rooted phylogenetic tree is a network, but that networks are more general since they can represent evolutionary events where species combine rather than speciate.…”

Reconstructing Phylogenetic Level-1 Networks from Nondense Binet and Trinet Sets

“…5 The set T of trinets that we use to illustrate the inner workings of our algorithm Although we could do Steps 2 and 3 in a purely brute force way, we present several structural lemmas which restrict the search space and will be useful in Sect. 5.…”

“…[10,18,19,22]). Even so, it has been observed that the set of triplets displayed by a level-1 network does not necessarily provide all of the information required to uniquely define or encode the network [5]. Motivated by this observation, in [11] an algorithm was developed for constructing level-1 networks from a network analogue of triplets: rooted binary networks with three leaves, or trinets.…”

Spaces of phylogenetic networks from generalized nearest-neighbor interchange operations

“…This is a new metric on rooted level-1 phylogenetic networks based on local operations, and it would be interesting to compare it with other known metrics on such networks (see, e.g. Cardona et al, 2011;Gambette and Huber, 2012;Huber and Moulton, 2013;Nakhleh, 2010, and the references therein). Finally, we remark here that it is possible to work out a formula to compute the size of the d rLST neighborhood of a rooted level-1 network via an approach similar to the proof of Theorem 3, but we shall not present this here as it is quite involved.…”