Bounding the Size of a Network Defined By Visibility Property

Phylogenetic networks are mathematical structures for modeling and visualization of reticulation processes in the study of evolution. Galled networks, reticulation visible networks, nearly-stable networks and stable-child networks are the four classes of phylogenetic networks that are recently introduced to study the topological and algorithmic aspects of phylogenetic networks. We prove the following results. (1) A binary galled network with n leaves has at most 2(n-1) reticulation nodes. (2) A binary nearly-stable network with n leaves has at most 3(n-1) reticulation nodes. (3) A binary stable-child network with n leaves has at most 7(n-1) reticulation nodes.Comment: 23 pages, 9 figure

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

The compressions of reticulation-visible networks are tree-child

Rooted phylogenetic networks are rooted acyclic digraphs. They are used to model complex evolution where hybridization, recombination and other reticulation events play important roles. A rigorous definition of network compression is introduced on the basis of the recent studies of the relationships between cluster, tree and rooted phylogenetic network. The concept reveals another interesting connection between the two well-studied network classes|tree-child networks and reticulation-visible networks|and enables us to define a new class of networks for which the cluster containment problem has a linear-time algorithm.Comment: 18 pages, 4 figure

On the Subnet Prune and Regraft Distance

Phylogenetic networks are rooted directed acyclic graphs that represent evolutionary relationships between species whose past includes reticulation events such as hybridisation and horizontal gene transfer. To search the space of phylogenetic networks, the popular tree rearrangement operation rooted subtree prune and regraft (rSPR) was recently generalised to phylogenetic networks. This new operation - called subnet prune and regraft (SNPR) - induces a metric on the space of all phylogenetic networks as well as on several widely-used network classes. In this paper, we investigate several problems that arise in the context of computing the SNPR-distance. For a phylogenetic tree $T$ and a phylogenetic network $N$, we show how this distance can be computed by considering the set of trees that are embedded in $N$ and then use this result to characterise the SNPR-distance between $T$ and $N$ in terms of agreement forests. Furthermore, we analyse properties of shortest SNPR-sequences between two phylogenetic networks $N$ and $N'$, and answer the question whether or not any of the classes of tree-child, reticulation-visible, or tree-based networks isometrically embeds into the class of all phylogenetic networks under SNPR

Counting Tree-Child Networks and Their Subclasses

Galled trees are studied as a recombination model in population genetics. This class of phylogenetic networks is generalized into tree-child, galled and reticulation-visible network classes by relaxing a structural condition imposed on galled trees. We count tree-child networks through enumerating their component graphs. Explicit counting formulas are also given for galled trees through their relationship to ordered trees, phylogenetic networks with few reticulations and phylogenetic networks in which the child of each reticulation is a leaf.Comment: 24 pages, 2 tables and 9 figure

Locating a Tree in a Reticulation-Visible Network in Cubic Time

In this work, we answer an open problem in the study of phylogenetic networks. Phylogenetic trees are rooted binary trees in which all edges are directed away from the root, whereas phylogenetic networks are rooted acyclic digraphs. For the purpose of evolutionary model validation, biologists often want to know whether or not a phylogenetic tree is contained in a phylogenetic network. The tree containment problem is NP-complete even for very restricted classes of networks such as tree-sibling phylogenetic networks. We prove that this problem is solvable in cubic time for stable phylogenetic networks. A linear time algorithm is also presented for the cluster containment problem.Comment: 25 pages, 3 figure

On Tree Based Phylogenetic Networks

A large class of phylogenetic networks can be obtained from trees by the addition of horizontal edges between the tree edges. These networks are called tree based networks. Reticulation-visible networks and child-sibling networks are all tree based. In this work, we present a simply necessary and sufficient condition for tree-based networks and prove that there is a universal tree based network for each set of species such that every phylogenetic tree on the same species is a base of this network. The existence of universal tree based network implies that for any given set of phylogenetic trees (resp. clusters) on the same species there exists a tree base network that display all of them.Comment: 17 pages, 6 figure

The SNPR neighbourhood of tree-child networks

Network rearrangement operations like SNPR (SubNet Prune and Regraft), a recent generalisation of rSPR (rooted Subtree Prune and Regraft), induce a metric on phylogenetic networks. To search the space of these networks one important property of these metrics is the sizes of the neighbourhoods, that is, the number of networks reachable by exactly one operation from a given network. In this paper, we present exact expressions for the SNPR neighbourhood of tree-child networks, which depend on both the size and the topology of a network. We furthermore give upper and lower bounds for the minimum and maximum size of such a neighbourhood.Comment: update to published version and fix typo

Displaying trees across two phylogenetic networks

Phylogenetic networks are a generalization of phylogenetic trees to leaf-labeled directed acyclic graphs that represent ancestral relationships between species whose past includes non-tree-like events such as hybridization and horizontal gene transfer. Indeed, each phylogenetic network embeds a collection of phylogenetic trees. Referring to the collection of trees that a given phylogenetic network $N$ embeds as the display set of $N$, several questions in the context of the display set of $N$ have recently been analyzed. For example, the widely studied Tree-Containment problem asks if a given phylogenetic tree is contained in the display set of a given network. The focus of this paper are two questions that naturally arise in comparing the display sets of two phylogenetic networks. First, we analyze the problem of deciding if the display sets of two phylogenetic networks have a tree in common. Surprisingly, this problem turns out to be NP-complete even for two temporal normal networks. Second, we investigate the question of whether or not the display sets of two phylogenetic networks are equal. While we recently showed that this problem is polynomial-time solvable for a normal and a tree-child network, it is computationally hard in the general case. In establishing hardness, we show that the problem is contained in the second level of the polynomial-time hierarchy. Specifically, it is $\Pi_2^P$-complete. Along the way, we show that two other problems are also $\Pi_2^P$-complete, one of which being a generalization of Tree-Containment

Tree-based networks: characterisations, metrics, and support trees

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer. One way to obtain such a network is by starting with a (rooted) phylogenetic tree $T$, called a base tree, and adding arcs between arcs of $T$. The class of phylogenetic networks that can be obtained in this way is called tree-based networks and includes the prominent classes of tree-child and reticulation-visible networks. Initially defined for binary phylogenetic networks, tree-based networks naturally extend to arbitrary phylogenetic networks. In this paper, we generalise recent tree-based characterisations and associated proximity measures for binary phylogenetic networks to arbitrary phylogenetic networks. These characterisations are in terms of matchings in bipartite graphs, path partitions, and antichains. Some of the generalisations are straightforward to establish using the original approach, while others require a very different approach. Furthermore, for an arbitrary tree-based network $N$, we characterise the support trees of $N$, that is, the tree-based embeddings of $N$. We use this characterisation to give an explicit formula for the number of support trees of $N$ when $N$ is binary. This formula is written in terms of the components of a bipartite graph.Comment: 20 pages, 6 figure