Totally Split-Decomposable Metrics of Combinatorial Dimension Two

Abstract.

The combinatorial dimension of a metric space (X; d), denoted by dim_combin(d), arises naturally in the subject of T-theory, and, in case X is finite, corresponds with the (topological) dimension of the tight span associated to d. Metric spaces of combinatorial dimension at most one are well understood; they are precisely the metric spaces that can be embedded into $\Bbb R$-trees. However, the structure of metric spaces of higher dimension is not so well understood. Indeed, even 2-dimensional finite metric spaces can have a rich structure.¶ In this paper, we study finite metric spaces of combinatorial dimension two that are, in addition, totally split decomposable. In particular, we give several characterizations of such metrics derived through the study of a certain map that relates the tight span of a totally split-decomposable metric with its Buneman complex.