4 The diagram represents a system of pipes through which fluid can flow from two sources, \(S _ { 1 }\) and \(S _ { 2 }\), to a sink, \(T\). Most of the pipes have valves which restrict the flow to one direction only. However, the flow in arc \(D E\) can be in either direction. The weights on the arcs show the lower capacities and the upper capacities of the pipes in litres per second.
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1. Add a supersource, \(S\), to the copy of the diagram in the answer book, and weight the arcs attached to it with appropriate lower and upper capacities.
2. The cut \(\alpha\) partitions the vertices into the sets \(\left\{ S , S _ { 1 } , S _ { 2 } , A , C \right\} , \{ B , D , E , T \}\). By considering the cut arcs only, calculate the maximum and minimum capacities of cut \(\alpha\).
3. Show that the maximum capacity of the cut \(\left\{ S , S _ { 1 } , S _ { 2 } , A , E \right\} , \{ B , C , D , T \}\) is 47 litres per second. A flow is set up in which the arcs \(S _ { 1 } A , S _ { 1 } B , S _ { 2 } C , A E , C E\) and \(D T\) are all at their lower capacities.
4. Show the flow in each arc on the diagram in the answer book, indicating the direction of the flow in \(\operatorname { arc } D E\).
5. What is the maximum amount, in litres per second, by which the flow can be augmented using the routes \(S _ { 1 } A D T\) and \(S _ { 2 } C E T\) ?
6. Find the maximum possible flow through the system, explaining how you know both that this is feasible and that it cannot be exceeded.