4 Answer parts (v) and (vi) of this question on the insert provided. The diagram represents a system of pipes through which fluid can flow. The weights on the arcs show the lower and upper capacities of the pipes in litres per second.
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1. Which vertex is the source and which vertex is the sink?
2. Cut \(\alpha\) partitions the vertices into the sets \(\{ A , B , C \} , \{ D , E , F , G , H , I \}\). Calculate the capacity of cut \(\alpha\).
3. Explain why partitioning the vertices into sets \(\{ A , D , G \} , \{ B , C , E , F , H , I \}\) does not give a cut.
4. (a) How many litres per second must flow along arc \(D G\) ?
   (b) Explain why the arc \(A D\) must be at its upper capacity. Hence find the flow in \(\operatorname { arc } B A\).
   (c) Explain why at least 7 litres per second must flow along arc \(B C\).
5. Use the diagrams in the insert to show a minimum feasible flow and a maximum feasible flow. The upper capacity of \(B C\) is now increased from 8 to 18 .
6. (a) Use the diagram in the insert to show a flow of 19 litres per second.
   (b) List the saturated arcs when 19 litres per second flows through the network. Hence, or otherwise, find a cut of capacity 19 .
7. Explain how your answers to part (vi) show that 19 litres per second is the maximum flow.