Binets: Fundamental Building Blocks for Phylogenetic Networks

Abstract

Phylogenetic networks are a generalization of evolutionary trees that are used by biologists to represent the evolution of organisms which have undergone reticulate evolution. Essentially, a phylogenetic network is a directed acyclic graph having a unique root in which the leaves are labelled by a given set of species. Recently, some approaches have been developed to construct phylogenetic networks from collections of networks on 2- and 3-leaved networks, which are known as binets and trinets, respectively. Here we study in more depth properties of collections of binets, one of the simplest possible types of networks into which a phylogenetic network can be decomposed. More specifically, we show that if a collection of level-1 binets is compatible with some binary network, then it is also compatible with a binary level-1 network. Our proofs are based on useful structural results concerning lowest stable ancestors in networks. In addition, we show that, although the binets do not determine the topology of the network, they do determine the number of reticulations in the network, which is one of its most important parameters. We also consider algorithmic questions concerning binets. We show that deciding whether an arbitrary set of binets is compatible with some network is at least as hard as the well-known graph isomorphism problem. However, if we restrict to level-1 binets, it is possible to decide in polynomial time whether there exists a binary network that displays all the binets. We also show that to find a network that displays a maximum number of the binets is NP-hard, but that there exists a simple polynomial-time 1/3-approximation algorithm for this problem. It is hoped that these results will eventually assist in the development of new methods for constructing phylogenetic networks from collections of smaller networks.

Combining Networks Using Cherry Picking Sequences

“…. , 1 do Subdivide the incoming edge of xi with a node px; 8 Subdivide the incoming edge of yi with a node py; 9 Add the edge pypx to N ; Subdivide the incoming edge of yi with a node py; 12 Add a new node labelled xi to N ; 13 Add the edge pyxi to N ; 14 return N ;…”

“…Instead of having almost all leaves, the subnetworks can also be allowed to have only few leaves. For example, low level networks can be reconstructed from their full set of binets [5,12], trinets [10,16] or quarnets [4].…”

Reconstructing Tree-Child Networks from Reticulate-Edge-Deleted Subnetworks

“…Building blocks are generally some class of subnetworks, e.g., binets (van Iersel et al. 2017), trinets (Huber and Moulton 2013), or trees, used to infer the original network. A potential problem here is that there could be more than one network with the same building blocks.…”

“…2017; van Iersel et al. 2017). Two of the more prominent network classes are the tree-child networks (Cardona et al.…”

Caterpillars on three and four leaves are sufficient to reconstruct normal networks

“…Partly due to the aforementioned negative deterministic results, recent studies have investigated whether or not rooted phylogenetic networks are determined by their embedded subnetworks like binets [10,14] and trinets [10,12,13], that is, rooted phylogenetic networks on two and three leaves, respectively. It has been shown that trinets determine rooted binary level-two and rooted binary tree-child networks [13].…”