Sufficient conditions for realizability of vertex-weighted directed graphs

In this paper we present several sufficient conditions for realizability of terminal capacity matrices of vertex-weighted directed graphs. First necessary and sufficient conditions for a matrix to be realizable by a vertex-weighted directed tree are derived and an algorithm for synthesis is given. Next, as an intermediate step, necessary and sufficient conditions for realizability by a vertex-weighted undirected tree are obtained. These results are then combined to give necessary and sufficient conditions for realizability as a vertex-weighted tree containing both directed and undirected branches. As a corollary, we obtain necessary and sufficient conditions for realizability as a vertex-weighted tree that is a directed path. A simple extension of this result gives realizability conditions for a vertex-weighted directed circuit. Finally we develop necessary and sufficient conditions for realizability as a vertex-weighted graph containing no directed circuits.