New characterisations of tree-based networks and proximity measures

Abstract

Phylogenetic networks are a type of directed acyclic graph that represent how a set X of present-day species are descended from a common ancestor by processes of speciation and reticulate evolution. In the absence of reticulate evolution, such networks are simply phylogenetic (evolutionary) trees. Moreover, phylogenetic networks that are not trees can sometimes be represented as phylogenetic trees with additional directed edges placed between their edges. Such networks are called tree-based, and the class of phylogenetic networks that are tree-based has recently been characterised. In this paper, we establish a number of new characterisations of tree-based networks in terms of path partitions and antichains (in the spirit of Dilworth's theorem), as well as via matchings in a bipartite graph. We also show that a temporal network is treebased if and only if it satisfies an antichain-to-leaf condition. In the second part of the paper, we define three indices that measure the extent to which an arbitrary phylogenetic network deviates from being tree-based. We describe how * Corresponding author.E-mail addresses: a.francis@westernsydney.edu.au (A. Francis), charles.semple@canterbury.ac.nz (C. Semple), mike.steel@canterbury.ac.nz (M. Steel).

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

“…Throughout this manuscript, we will assume that |X| ≥ 2, and that N is proper. A proper network is one for which all components obtained by removing a cut edge or cut vertex contain at least one element of X (following the definition of [2], more general than the one given in [5]). Note that a network that is not proper cannot be tree-based (cf.…”

“…Figure 6), while the relationship between δ NNI (N ) and (N ), or λ(N ) and e(N ), is not known. N r is tree-based if it can be constructed from a tree by additional arcs that avoid cycles, and this was shown to be equivalent to the statement that N r has a spanning tree whose leaves are those of X [6]: this formulation was used as the definition of tree-based in [5]. The first of these definitions was generalized to the non-binary case in several ways in [10], the main one of which is as follows: Definition 4, [10]).…”

“…Tree-based proximity, rooted and unrooted. Proximity measures for rooted phylogenetic networks have already been well-studied [5,12], but an interesting question is whether the new measures for unrooted phylogenetic networks introduced in the present paper have analogs in the rooted setting.…”

“…Another interesting aspect about the difference between rooted and unrooted networks is that the proximity measures based on bipartite matchings introduced for rooted networks in [5] do not carry over to the unrooted case. This is due to the fact that finding a maximum matching and determining its size can be done in polynomial time.…”

Tree-based unrooted nonbinary phylogenetic networks

“…As unrooted nonbinary tree-based networks have not yet been heavily studied, there are a number of interesting avenues for further research. For instance, given the wide variety of characterisations for tree-based networks in the binary and nonbinary rooted settings [6,7,9,13,11], the following natural question arises: Question 6.1. Are there analogous characterisations for tree-based networks for the unrooted nonbinary case, especially computationally efficient ones?…”