Consider the following network (the numbers associated with the edges are the edge capacities).

a. Find the maximum flow f and a minimum cut.

b. Draw the final residual graph Gf (along with its edge capacities). In this residual network, mark the vertices reachable from S and the vertices from which T is reachable.

c. An edge of a network is called a bottleneck edge if increasing its capacity results in an increase in the maximum flow. List all bottleneck edges in the above network.

d. Give a very simple example (containing at most four vertices) of a network which has no bottleneck edges.

e. Give an efficient algorithm to identify all bottleneck edges in a network. (Hint: Execute a known network flow algorithm (e.g., Ford-Fulkerson), and then examine the residual graph.)

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