5. \begin{figure}[h] \end{figure} Figure 1 shows a capacitated, directed network. 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 corresponding pipes, in litres per second.

1. Calculate the capacity of
   1. cut \(C _ { 1 }\)
   2. cut \(C _ { 2 }\)
2. Using only the capacities of cuts \(C _ { 1 }\) and \(C _ { 2 }\), state what can be deduced about the maximum flow through the system. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{262aa0e6-479f-447a-94db-aeb901b3c6fe-6_775_1516_169_278} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows an initial flow through the same network.
3. State the value of the initial flow.
4. By entering values along \(B C , C F\) and \(D T\), complete the labelling procedure on Diagram 1 in the answer book.
5. Use the labelling procedure to find a maximum flow through the network. You must list each flow-augmenting route you use, together with its flow.
6. Use your answer to (e) to find a maximum flow pattern for this system of pipes and draw it on Diagram 2 in the answer book.
7. Prove that the answer to (f) is optimal. A vertex restriction is now applied to \(B\) so that no more than 16 litres per second can flow through it.
8. 1. Complete Diagram 3 in the answer book to show this restriction.
   2. State the value of the maximum flow with this restriction.