# Level-1 Phylogenetic Networks and their Balanced Minimum Evolution Polytopes

## TL;DR

This paper introduces polytopes associated with level-1 phylogenetic networks, enabling linear programming methods to reconstruct such networks and revealing their geometric structure and relation to existing polytopes.

## Contribution

It defines a new family of polytopes for phylogenetic networks, connecting them to BME and TSP polytopes, and characterizes their vertices and facets.

## Key findings

- Vertices correspond to level-1 phylogenetic networks.
- Polytopes include BME and TSP polytopes as special cases.
- Facets correspond to splits and minimal facets.

## 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.