A$4$C $972$ ST $265$ B$3$D $($a$)$ Find the maximum flow f and a minimum cut. $($b$)$ DrawtheresidualgraphGf $($alongwithitsedgecapacities$).$Inthisresidualnetwork,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 nodes$)$ of a network which has no bottleneck edges. $($e$)$ Give an efficient algorithm to identify all bottleneck edges in a network. $($Hint: Start by running the usual network flow algorithm, and then examine the residual graph.$)$