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On graphlike k-dissimilarity vectors

Let {\cal G}=(G,w) be a positive-weighted simple finite graph, that is, let G be a simple finite graph endowed with a function w from the set of the edges of G to the set of the 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 i_1,..., i_k vertices of G, let D_{i_1,.... i_k}({\cal G}) be the minimum of the weights of the subgraphs of G connecting i_1,..., i_k. The D_{i_1,.... i_k}({\cal G}) are called k-weights of {\cal G}. Given a family of positive real numbers parametrized by the k-subsets of {1,..., n}, {D_I}_{I k-subset of {1,...,n}}, we can wonder when there exist a weighted graph {\cal G} (or a weighted tree) and an n-subset {1,..., n} of the set of its vertices such that D_I({\cal G}) =D_I for any I k-subset of {1,...,n}. In this paper we study this problem in the case k=n-1.