On the structure of the tight-span of a totally split-decomposable metric

Authors:
K. T. Huber;J. H. Koolen;V. Moulton
Affiliations:
School of Computing Sciences, University of East Anglia, Norwich, UK;Division of Applied Mathematics KAIST, Kusongdong, Yusongku, Daejon, Republic of Korea;School of Computing Sciences, University of East Anglia, Norwich, UK
Venue:
European Journal of Combinatorics
Year:
2006

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Abstract

The tight-span of a finite metric space is a polytopal complex with a structure that reflects properties of the metric. In this paper we consider the tight-span of a totally split-decomposable metric. Such metrics are used in the field of phylogenetic analysis, and a better knowledge of the structure of their tight-spans should ultimately provide improved phylogenetic techniques. Here we prove that a totally split-decomposable metric is cell-decomposable. This allows us to break up the tight-span of a totally split-decomposable metric into smaller, easier to understand tight-spans. As a consequence we prove that the cells in the tight-span of a totally split-decomposable metric are zonotopes that are polytope isomorphic to either hypercubes or rhombic dodecahedra.