Realization of a network from the upper and lower bounds of the distances (or capacities) between vertices

In graph and network theory, an important problem to find networks satisfying given conditions. If distances (or capacities) between vertices for every vertex pair are given, the necessary and sufficient conditions for existence of networks satisfying given conditions are known. The authors give the upper and lower bounds of distances (or capacities) between vertices for every vertex pair. They obtain necessary and sufficient conditions for existence of networks satisfying the given conditions. These are a generalization of the results by S. L. Hakimi et al. and R. E. Gomory et al. If N is a network satisfying the given conditions, the strict upper and lower bounds of distances (or capacities) between vertices in N are given