Flow networks: Push flow on either edges but not both!
You can model the constraint of not using two edges (u,v) and (u,w) of capacity one simultaneously in a flow network by introducing a vertex u' and an edge (u',u) with capacity one, but I'm not aware of a more general way to do this without extending the concept of a flow network.

If your idea for reducing the maximum independent set problem to a network flow problem worked, you'd have the basis for a proof that P=NP, which is extremely unlikely.