Extremal optimal realizations

An optimal realization of a metric d on a set X is a weighted graph G = (V, E, w) such that X ⊆ V , dG(x, y) = d(x, y) for all x ∈ X, and ∑ e∈E w(e) is minimal. In this paper, we consider the conjecture that any finite metric (X, d) has some optimal realization obtainable as a subgraph of its tight span. For an extremal optimal realization G, defined as an optimal realizations for which the number of geodesics is maximal, we show that the contraction of G into T (X, d) is injective. Moreover, if T (X, d) is of dimension at most two then the above conjecture holds.