Tree-Based Unrooted Phylogenetic Networks

Abstract

Phylogenetic networks are a generalization of phylogenetic trees that are used to represent non-tree-like evolutionary histories that arise in organisms such as plants and bacteria, or uncertainty in evolutionary histories. An unrooted phylogenetic network on a non-empty, finite set X of taxa, or network, is a connected, simple graph in which every vertex has degree 1 or 3 and whose leaf set is X . It is called a phylogenetic tree if the underlying graph is a tree. In this paper we consider properties of tree-based networks, that is, networks that can be constructed by adding edges into a phylogenetic tree. We show that although they have some properties in common with their rooted analogues which have recently drawn much attention in the literature, they have some striking differences in terms of both their structural and computational properties. We expect that our results could eventually have applications to, for example, detecting horizontal gene transfer or hybridization which are important factors in the evolution of many organisms.

Tree-metrizable HGT networks

“…The resulting phylogenetic network is referred to as a minimal support network of B in N, denoted N(B), and is unique up to rearrangement of the leaf labels. We note that minimal support networks are perhaps most easily understood as the rooted equivalent of B N from [4]. We also note here that if a minimal support network of a biconnected component in a network N is not tree-metrizable, it easily follows that N is not tree-metrizable.…”

“…A 1 + A 2 = α(a 1 + a 2 + a 3 ) + (1 − α)(a 1 + a 2 + a 3 ) (distance v 1 to v 2 ) = a 1 + a 2 + a 3 A 1 + A 3 = a 1 + a 5 + αa 2 + (1 − α)a 4 (distance v 1 to v 3 ) A 2 + A 3 = a 3 + a 5 + (1 − α)(a 2 + a 4 ).…”

How tree-based is my network? Proximity measures for unrooted phylogenetic networks

“…If N contains such a support tree T , it is called tree-based. This is a direct generalization of the definition for binary phylogenetic networks in [3], and coincides with the notion of "loosely" tree-based introduced in [7]. Note that T is not necessarily a phylogenetic tree as it may contain degree-2 vertices.…”

“…It can be easily shown that, while the rooted counterparts of , t, and p introduced in [5] can all be calculated in polynomial time, this is not possible for the above unrooted proximity measures, because otherwise, it could easily be decided if they equal 0 (which would contradict the known NP-completeness of the unrooted treebasedness decision problem [3], cf. Section 7).…”

“…There is a spanning tree for the network that consists of the Hamiltonian path plus the additional edges to the leaves from N \ LCUT (N ) or plus the additional (i) (ii) Figure 1. (i) Level 5 non-tree-based network whose vertices are covered by two edge-disjoint paths, shown in bold (this example of a non-tree-based network was independently presented in [2] and the erratum [4] to [3]). (ii) The same network with an additional diagonal edge making it tree-based (support tree highlighted in bold).…”