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Area of Science:

  • Phylogenetic Theory
  • Computational Biology
  • Discrete Geometry

Background:

  • The structure of the balanced minimal evolution (BME) polytope is crucial for understanding phylogenetic relationships.
  • Identifying the top-dimensional facets of this polytope is a key challenge in phylogenetic theory.
  • Previous work by Hodge, Haws, and Yoshida established a lattice of faces for the BME polytope.

Purpose of the Study:

  • To investigate the face structure of the balanced minimal evolution (BME) polytope.
  • To identify and characterize the top-dimensional facets of the BME polytope.
  • To establish a connection between phylogenetic tree splits and the BME polytope's combinatorial structure.

Main Methods:

  • Mathematical analysis of the balanced minimal evolution (BME) polytope.
  • Exploration of the poset of faces of the BME polytope.
  • Comparison with the structure of the permutoassociahedron and its quotients.

Main Results:

  • A sublattice within the BME polytope's face poset was identified, isomorphic to a quotient of the permutoassociahedron.
  • This sublattice represents compatible sets of splits from phylogenetic trees.
  • Maximal elements in this new poset correspond to single leaf splits, with most being facets of the BME polytope, exhibiting exponential growth.

Conclusions:

  • The study reveals a novel combinatorial structure within the BME polytope related to phylogenetic splits.
  • This extends previous findings on the BME polytope's face lattice.
  • The results provide new insights into the complexity and structure of phylogenetic models.