Level-1 phylogenetic networks and their balanced minimum evolution polytopes

Abstract

Balanced minimum evolution is a distance-based criterion for the reconstruction of phylogenetic trees. Several algorithms exist to find the optimal tree with respect to this criterion. One approach is to minimize a certain linear functional over an appropriate polytope. Here we present polytopes that allow a similar linear programming approach to finding phylogenetic networks. We investigate a two-parameter family of polytopes that arise from phylogenetic networks, and which specialize to the Balanced Minimum Evolution polytopes as well as the Symmetric Travelling Salesman polytopes. We show that the vertices correspond to certain level-1 phylogenetic networks, and that there are facets or faces for every split. We also describe minimal facets and a family of faces for every dimension.2000 Mathematics Subject Classification. 90C05, 52B11, 92D15.

Phylogenetic Networks as Circuits With Resistance Distance

“…Therefore, as a measured set of pairwise distances approach the resistance distance of N, the output of neighbor-net will approach the faithfully phylogenetic circular split network N. This is in contrast to minimum path distance where some genetic connections are assumed to be negligible, and then are lost in the output of neighbor-net. However, the theorems about minimum path distance, specifically Theorems 8, 9, and 11 of Durell and Forcey (2020), play an important role in the proof of Theorem 4.5 here. Here we repeat some of the same introductory definitions and remarks and then extend the results to resistance distance.…”

“…This function is shown to exist in Gambette et al (2017), and described on the split networks which are images of the function . In Durell and Forcey (2020) and Forcey and Scalzo (2020a), we define the general function L as follows:…”

“…We next define functions between the weighted split networks and the weighted phylogenetic networks. As previously explained in Durell and Forcey (2020) and Forcey and Scalzo (2020a), we begin by extending the function L to a weighted version L w .…”

“…In Durell and Forcey (2020), the authors describe a new family of polytopes. This family lies between the Symmetric Traveling Salesman Polytope [STSP(n)] and the Balanced Minimum Evolution Polytope [BME(n)].…”

“…The convex hull of all the x(N) such that binary N has n leaves and k nontrivial bridges is the level-1 network polytope BME(n, k). As shown in Durell and Forcey (2020), the vertices of BME(n, k) are precisely the vectors x(N) for N binary with n leaves and k nontrivial bridges. In light of Theorems 3.1 and 3.2, we can now characterize the vertices in terms of resistance distance: Theorem 4.2.…”