The Buneman index via polyhedral split decomposition

The split decomposition of a metric using the so-called isolation index was established by Bandelt and Dress, and it is a theoretical foundation for certain phylogenetic network reconstruction methods. Recently, Hirai gave a geometric interpretation to the split decomposition, and provided an extension of the split decomposition for a distance, i.e., a symmetric nonnegative function with zero diagonal. By Herrmann and Moulton, such a geometric approach was further applied to polyhedral functions, called tight-spans in their style, defined on various vector configurations. This paper addresses the split decomposition with respect to a particular kind of vector configuration, which does not satisfy the assumption imposed by Hirai or Herrmann and Moulton. As a result, we obtain geometrically the Buneman index, which is also used to construct a phylogenetic tree. Moreover, this paper deals with the combinatorial aspect of the polyhedral split decomposition, and gives a combinatorial characterization of the split-decomposability with the aid of the matroid associated with the vector configuration.