In evolutionary biology, phylogenetic networks are graphs that provide a flexible framework for representing complex evolutionary histories that involve reticulate evolutionary events. Recently, phylogenetic studies have started to focus on a special class of such networks called semi-directed networks. These graphs are defined as mixed graphs that can be obtained by de-orienting some of the arcs in some rooted phylogenetic network, that is, a directed acyclic graph whose leaves correspond to a collection of species and that has a single source or root vertex. However, this definition of semi-directed networks is implicit in nature since it is not clear when a mixed-graph enjoys this property or not. In this paper, we introduce novel, explicit mathematical characterizations of semi-directed networks, and also multi-semi-directed networks, that is mixed graphs that can be obtained from directed phylogenetic networks that may have more than one root. In addition, through extending foundational tools from the theory of rooted networks into the semi-directed setting—such as cherry picking sequences, omnians, and path partitions—we characterize when a (multi-)semi-directed network can be obtained by de-orienting some rooted network that is contained in one of the well-known classes of tree-child, orchard, tree-based or forest-based networks. These results address structural aspects of (multi-)semi-directed networks and pave the way to improved theoretical and computational analyses of such networks, for example, within the development of algebraic evolutionary models that are based on such networks.

With the increasing availability of genomic data, biologists aim to find more accurate descriptions of evolutionary histories influenced by secondary contact, where diverging lineages reconnect before diverging again. Such reticulate evolutionary events can be more accurately represented in phylogenetic networks than in phylogenetic trees. Since the root location of phylogenetic networks cannot be inferred from biological data under several evolutionary models, we consider semi-directed (phylogenetic) networks: partially directed graphs without a root in which the directed edges represent reticulate evolutionary events. By specifying a known outgroup, the rooted topology can be recovered from such networks. We introduce the algorithm Squirrel (Semi-directed Quarnet-based Inference to Reconstruct Level-1 Networks) which constructs a semi-directed level-1 network from a full set of quarnets (four-leaf semi-directed networks). Our method also includes a heuristic to construct such a quarnet set directly from sequence alignments. We demonstrate Squirrel’s performance through simulations and on real sequence data sets, the largest of which contains 29 aligned sequences close to 1.7 Mb long. The resulting networks are obtained on a standard laptop within a few minutes. Lastly, we prove that Squirrel is combinatorially consistent: given a full set of quarnets coming from a triangle-free semi-directed level-1 network, it is guaranteed to reconstruct the original network. Squirrel is implemented in Python, has an easy-to-use graphical user interface that takes sequence alignments or quarnets as input, and is freely available [...]

In evolutionary biology, networks are becoming increasingly used to represent evolutionary histories for species that have undergone non-treelike or reticulate evolution. Such networks are essentially directed acyclic graphs with a leaf set that corresponds to a collection of species, and in which non-leaf vertices with indegree 1 correspond to speciation events and vertices with indegree greater than 1 correspond to reticulate events such as gene transfer. Recently forest-based networks have been introduced, which are essentially (multi-rooted) networks that can be formed by adding some arcs to a collection of phylogenetic trees (or phylogenetic forest), where each arc is added in such a way that its ends always lie in two different trees in the forest. In this paper, we consider the complexity of deciding whether a given network is proper forest-based, that is, whether it can be formed by adding arcs to some underlying phylogenetic forest which contains the same number of trees as there are roots in the network. More specifically, we show that it is NP-complete to decide whether a tree-child network with m roots is proper forest-based, for each m≥2. Moreover, for binary networks the problem remains NP-complete when m≥3 but becomes polynomial-time solvable for m=2. We also give a fixed parameter tractable (FPT) algorithm, with parameters the maximum outdegree of a vertex, the number of roots, and the number of indegree 2 vertices, for deciding if a semi-binary network is proper forest-based. A key element in proving our results is a new characterization for when a network with m roots is proper forest-based in terms of certain m-colorings.

Phylogenetic networks are graphs that are used to represent evolutionary relationships between different taxa. They generalize phylogenetic trees since for example, unlike trees, they permit lineages to combine. Recently, there has been rising interest in semi-directed phylogenetic networks, which are mixed graphs in which certain lineage combination events are represented by directed edges coming together, whereas the remaining edges are left undirected. One reason to consider such networks is that it can be difficult to root a network using real data. In this paper, we consider the problem of when a semi-directed phylogenetic network is defined or encoded by the smaller networks that it induces on the 4-leaf subsets of its leaf set. These smaller networks are called quarnets. We prove that semi-directed binary level-2 phylogenetic networks are encoded by their quarnets, but that this is not the case for level-3. In addition, we prove that the so-called blob tree of a semi-directed binary network, a tree that gives the coarse-grained structure of the network, is always encoded by the quarnets of the network. These results are relevant for proving the statistical consistency of programs that are currently being developed for reconstructing phylogenetic networks from practical data, such as the recently developed Squirrel software tool.

This paper studies the relationship between undirected (unrooted) and directed (rooted) phylogenetic networks. We describe a polynomial-time algorithm for deciding whether an undirected nonbinary phylogenetic network, given the locations of the root and reticulation vertices, can be oriented as a directed nonbinary phylogenetic network. Moreover, we characterize when this is possible and show that, in such instances, the resulting directed nonbinary phylogenetic network is unique. In addition, without being given the location of the root and the reticulation vertices, we describe an algorithm for deciding whether an undirected binary phylogenetic network N can be oriented as a directed binary phylogenetic network of a certain class. The algorithm is fixed-parameter tractable (FPT) when the parameter is the level of N and is applicable to classes of directed phylogenetic networks that satisfy certain conditions. As an example, we show that the well-studied class of binary tree-child networks satisfies these conditions.

Recently it was shown that a certain class of phylogenetic networks, called level-2 networks, cannot be reconstructed from their associated distance matrices. In this paper, we show that they can be reconstructed from their induced shortest and longest distance matrices. That is, if two level-2 networks induce the same shortest and longest distance matrices, then they must be isomorphic. We further show that level-2 networks are reconstructible from their shortest distance matrices if and only if they do not contain a subgraph from a family of graphs. A generator of a network is the graph obtained by deleting all pendant subtrees and suppressing degree-2 vertices. We also show that networks with a leaf on every generator side are reconstructible from their induced shortest distance matrix.

The problem of realizing finite metric spaces in terms of weighted graphs has many applications. For example, the mathematical and computational properties of metrics that can be realized by trees have been well-studied and such research has laid the foundation of the reconstruction of phylogenetic trees from evolutionary distances. However, as trees may be too restrictive to accurately represent real-world data or phenomena, it is important to understand the relationship between more general graphs and distances. In this paper, we introduce a new type of metric called a cactus metric, that is, a metric that can be realized by a cactus graph. We show that, just as with tree metrics, a cactus metric has a unique optimal realization. In addition, we describe an algorithm that can recognize whether or not a metric is a cactus metric and, if so, compute its optimal realization in O(n³) time, where n is the number of points in the space.

Recently, much attention has been devoted to the construction of phylogenetic networks which generalize phylogenetic trees in order to accommodate complex evolutionary processes. Here, we present an efficient, practical algorithm for reconstructing level-1 phylogenetic networks-a type of network slightly more general than a phylogenetic tree-from triplets. Our algorithm has been made publicly available as the program Lev1athan. It combines ideas from several known theoretical algorithms for phylogenetic tree and network reconstruction with two novel subroutines. Namely, an exponential-time exact and a greedy algorithm both of which are of independent theoretical interest. Most importantly, Lev1athan runs in polynomial time and always constructs a level-1 network. If the data are consistent with a phylogenetic tree, then the algorithm constructs such a tree. Moreover, if the input triplet set is dense and, in addition, is fully consistent with some level-1 network, it will find such a network. The potential of Lev1athan is explored by means of an extensive simulation study and a biological data set. One of our conclusions is that Lev1athan is able to construct networks consistent with a high percentage of input triplets, even when these input triplets are affected by a low to moderate level of noise.