Uprooted Phylogenetic Networks

Abstract

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard model of a phylogenetic tree by also allowing for cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees or their unrooted versions, rooted phylogenetic networks are notoriously difficult to understand. To help alleviate this, recent work on them has also centered on their “uprooted” versions. By focusing on such graphs and the combinatorial concept of a split system which underpins an unrooted phylogenetic network, we show that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph (sometimes also called a median network) associated with the split system induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Along the way, we establish the 1-nested analogue of the fundamental “splits equivalence theorem” for phylogenetic trees and characterize maximal circular split systems.

Level-1 phylogenetic networks and their balanced minimum evolution polytopes

“…There is an even closer relationship between level-1 (and thus 1-nested) networks and circular split networks. In [8] it is shown that a split network s is circular if and only if there exists an unrooted level-1 network N such that s ⊂ Σ(N ). For instance the split network s in Figure 1 has splits a subset of those in Σ(N ) seen in Figure 2.…”

“…If s is an externally refined split network then L(s) is a binary level-1 phylogenetic network. Note that in [8] the authors define a similar function called N . Their function is equivalent to ours, if the definition in [8] is modified so as not to depend on k-marguerites.…”

Phylogenetic Networks as Circuits With Resistance Distance

“…Many of the definitions and notes here are repeated (sometimes verbatim) from Forcey and Scalzo (2020a) for the sake of self-containment. For further reference, see Steel (2016) and Gambette et al (2017).…”

“…Recall that a cycle in a graph is a path of edges that does not revisit any nodes except for the node at which it starts and ends. The following is defined in Gambette et al (2017): Definition 2.2. An unrooted phylogenetic network N is called 1nested when each edge of N is contained in at most one cycle, and N is triangle-free-all cycles are of length greater than 3 edges.…”

“…The definitions in this section are repeated from Forcey and Scalzo (2020a), but originate in Gambette et al (2017).…”