Let \(G\) be a finite strongly connected mixed graph (i.e., a graph with both undirected and directed edges, in which each vertex can be reached from every other vertex if directed edges can only be traversed in their direction of orientation). We establish a necessary and sufficient condition for it to be possible to transform some undirected edges of \(G\) into directed edges so that each vertex becomes the head of a prescribed number of newly directed edges and \(G\) remains strongly connected. A special case of this result yields a new proof (not requiring matroid techniques) of a necessary and sufficient condition for it to be possible to split each vertex of a finite connected graph into a prescribed number of vertices whilst preserving connectedness.

The Balanced Network Search (BNS) is an algorithm which finds a maximum balanced flow in a balanced network \({N}\). This algorithm is a way of using network flows to solve a number of standard problems, including maximum matchings, the factor problem, maximum capacitated \(b\)-matchings, etc., in general graphs. The value of a maximum balanced flow equals the capacity of a minimum balanced edge-cut. Flow-carrying balanced networks contain structures called generalized blossoms. They are not based on odd cycles. Rather they are the connected components of a residual sub-network of \({N}\). An algorithm is given for finding a maximum balanced flow, by constructing complementary pairs of valid augmenting paths.