Totally Split-Decomposable Metrics of
Combinatorial Dimension Two

A. Dress^{1}ㄛ K.T. Huber^{2}, and V. Moulton^{3}

dress@mathematik.uni-bielefeld.de

Annals of Combinatorics 5 (1) p.99-112 March, 2001

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 -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.

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.

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