Necessary and Sufficient Conditions for Realizability of Vertex-Weighted Communication Nets

A counter example is given to show that the conditions for a matrix to be realizable as the terminal-capacity matrix of a vertex-weighted communication net, which were given in 1962, are only necessary but not sufficient. Necessary and sufficient conditions for realizability are proposed and proved. Refinements in the description of a terminal-capacity matrix are presented and a new synthesis algorithm is introduced. Finally, conditions for special realizability classes of graphs are obtained as immediate consequences of the necessary and sufficient conditions.