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 i_1,..., i_k vertices of G, let D{i_1,..., i_k}(G) be the minimum of the weights of the subgraphs of G connecting i_1,..., i_k. The D{i_1,..., i_k}(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∈(_k~({1,..., n}))), 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 ∈ (_k~({1,...,n})). In this paper we study this problem in the case k = n?1.
展开▼
机译：令G =（G，w）是一个正加权的简单有限连通图，也就是说，让G是一个从G的边集到正实数的集合具有函数w的简单有限连通图。对于G的任何子图G'，我们将w（G'）定义为G'的边缘权重之和。对于G的任何i_1，...，i_k个顶点，令D {i_1，...，i_k}（G）为连接i_1，...，i_k的G的子图权重的最小值。 D {i_1，...，i_k}（G）称为G的k权重。给定一个由{1，...，n}，{D_I} _的k个子集参数化的正实数族（I∈（_k〜（{1，...，n}））），我们想知道何时存在加权图G（或加权树）和n个子集{1，...，n}它的顶点集合的任意一个，使得对于任何I∈（_k〜（{1，...，n}））D_I（G）= D_I。在本文中，我们研究了在k = n？1的情况下的问题。
展开▼
