Recent advances on the polyhedral combinatorics of the Balanced Minimum Evolution Problem (BMEP) enabled the characterization of a number of facets of its convex hull (also referred to as the BMEP polytope) as well as the discovery of connections between this polytope and the permutoassociahedron. In this article, we extend these studies, by presenting new results concerning some fundamental characteristics of the BMEP polytope, new facet-defining inequalities in the case of six or more taxa, a number of valid inequalities, and a polynomial time oracle to recognize its vertices. Our aim is to broaden understanding of the polyhedral combinatorics of the BMEP with a view to developing new and more effective exact solution algorithms.

平衡最小演化问题（BMEP）的多面体组合学的最新进展使得能够表征其凸包的多个小面（也称为BMEP多面体），并发现了该多面体与全黏合体之间的连接。在本文中，我们通过提供有关BMEP多态性的一些基本特征的新结果，在六个或更多类群的情况下新的刻面定义不等式，许多有效不等式以及用于识别其特征的多项式时间预言来扩展这些研究顶点。我们的目的是拓宽对BMEP的多面体组合的理解，以开发新的，更有效的精确求解算法。