Problem $3(20$ points$)$

Consider the following network $($the numbers are edge capacities$).$

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

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

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