This is called an augmenting path. The button run executes once this stage. Here is an open problem for Max Flow fans. Let G be a bipartite graph, where the left nodes are sources and the right nodes are sinks. Every source has an associated red and blue offer and every sink has an associated red and blue demand.
Every edge can transport exactly one red, one blue flow or no flow at all.
We say that the edge has integer capacity 1. We want to know if there is a flow satisfying all offers and demands, not exceeding edge capacities. It is NP-hard if G is an arbitrary bipartite graph.