Phylogenetic networks generalize evolutionary trees and are commonly used to represent evolutionary relationships between species that undergo reticulate processes such as hybridization, recombination and lateral gene transfer. Recently, there has been great interest in knowing which networks are determined or encoded by their trinets, that are rooted networks on three species. Van Iersel and Moulton showed that recoverable rooted binary level-2 phylogenetic networks are encoded by their trinets. Based on their work for level-2 networks, we show here that not all recoverable rooted binary level-3 networks are weakly encoded by their trinets, but most networks are. Further, although not all level-3 networks are weakly encoded by their trinets, we are able to prove that all recoverable rooted binary level-3 networks are encoded by their quarnets, that are rooted networks on four species.

Phylogenetic networks generalize evolutionary trees and are commonly used to represent evolutionary relationships between species that undergo reticulate evolutionary processes such as hybridization, recombination and lateral gene transfer. In this thesis all quarnets, networks on four species, of a network are assumed to be known. We prove that each recoverable undirected or semi-directed binary level-2 phylogenetic network without redundant biconnected components is encoded by its set of quarnets, meaning that the network is uniquely determined by its quarnets. Furthermore, two decomposition theorems for undirected and semi-directed binary phylogenetic networks are presented. These decomposition theorems are proved for undirected binary phylogenetic networks for all levels and for semi-directed binary phylogenetic networks that are at most level-2.

Evolutionary histories for species that cross with one another or exchange genetic material can be represented by leaf-labelled, directed graphs called phylogenetic networks. A major challenge in the burgeoning area of phylogenetic networks is to develop algorithms for building such networks by amalgamating small networks into a single large network. The level of a phylogenetic network is a measure of its deviation from being a tree; the higher the level of a network, the less treelike it becomes. Various algorithms have been developed for building level-1 networks from small networks. However, level-1 networks may not be able to capture the complexity of some data sets. In this paper, we present a polynomial-time algorithm for constructing a rooted binary level-2 phylogenetic network from a collection of 3-leaf networks or trinets. Moreover, we prove that the algorithm will correctly reconstruct such a network if it is given all of the trinets in the network as input. The algorithm runs in time O(t⋅n+n4) with t the number of input trinets and n the number of leaves. We also show that there is a fundamental obstruction to constructing level-3 networks from trinets, and so new approaches will need to be developed for constructing level-3 and higher level-networks.@en