On graphlike k-dissimilarity vectors

Let G = (G, w) be a positive-weighted simple finite connected graph, that is, let G be a simple finite connected graph endowed with a function w from the set of edges of G to the set of positive real numbers. For any subgraph G' of G, we define w(G') to be the sum of the weights of the edges of G'. For any i1, . . . , ik vertices of G, let D_{i1,..., ik}(G) be the minimum of the weights of the subgraphs of G connecting i1, . . . , ik. The D_{i1,..., ik}(G) are called k-weights of G. Given a family of positive real numbers parametrized by the k-subsets of {1, . . . , n}, {D_I}_I∈({1,..., n} \choose k ), we can wonder when there exist a weighted graph G (or a weighted tree) and an n-subset {1, . . . , n} of the set of its vertices such that D_I(G) = D_I for any I ∈ {1,...,n} \choose k . In this paper we study this problem in the case k = n−1.