Suppose that a maximum flow for G has been computed using Ford-Fulkerson, and a new edge with unit capacity is added to E. How the maximum flow can be efficiently updated. (t is not the value of the flow that must be updated, but the flow itself.
Let G' be the graph with the new edge e added to G. Note that we retain the capacity and flow for the remaining edges.
Now find an augmenting path p in G'.
If p exists, then update the flow along that path in G' by 1. Otherwise, the flow remains the same.
This gives the final flow values. This is correct because if p exists, then it passes through e. Hence, the flow update along p is exactly by 1. Since Folk-Fulkerson algorithm increases the flow in integral steps, there is no augmenting path in G' after this update.
If p doesn't exist, then by the mincut-maxflow argument, this is the maximum flow as the mincut is 0.