2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ddebe845-4280-471b-8da0-cb7211cea756-03_855_1820_210_127}
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\caption{Figure 1}
\end{figure}
An engineer monitors a system of pipes through which a fluid flows from the source, S , to the sink, T .
The engineer initialises the labelling procedure for this system, and the excess capacities and potential backflows are shown on the arrows either side of each arc, as shown in Figure 1.
- State the value of the initial flow.
- Obtain the capacity of the cut that passes through the arcs \(\mathrm { SA } , \mathrm { SB } , \mathrm { CE } , \mathrm { FE }\) and FJ .
- 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.
- Use your answer to (c) to draw a maximum flow pattern on Diagram 1 in the answer book.
- Prove that the answer to (d) is optimal.