Given a network as a directed acyclic graph with a source and sink, and non-negative edge capacities, I am interested in the extreme source-sink flows. More formally, I am interested in a characterization of the extreme points (vertices) of the capacitated flow polytope. Can someone provide me a reference to this problem or state the solution.
You can have a look at chapter 13, namely "Path and flow polyhedra and total unimodularity", of Alexander Schrijver's masterpiece "Combinatorial Optimization - Polyhedra and Efficiency". You will find characterizations based on cuts, which are the dual of flows in digraphs. This is linked to the famous max-flow min-cut theorem.