5 The network represents a system of pipes through which fluid can flow. The weights on the arcs show the lower and upper capacities for the pipes, in litres per second.
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1. Identify the source and explain how you know that the sink is at \(G\).
2. Calculate the capacity of the cut that separates \(\{ A , B , C , D , E , F \}\) from \(\{ G , H , I , J , K , L \}\).
3. Assuming that a feasible flow exists, explain why arc \(J G\) must be at its lower capacity. Write down the flows in arcs \(H K\) and \(I L\).
4. Assuming that a feasible flow exists, explain why arc HI must be at its upper capacity. Write down the flows in arcs \(E H\) and \(C B\).
5. Show a flow of 10 litres per second through the system.
6. Using your flows from part (v), label the arrows on the diagram to show the excess capacities and the potential backflows.
7. Write down a flow augmenting path from your diagram in part (vi), but do not update the excess capacities and the potential backflows. Hence show a maximum flow through the system, and state how you know that the flow is maximal.