7 Figure 2 shows a network of pipes. Water from two reservoirs, \(R _ { 1 }\) and \(R _ { 2 }\), is used to supply three towns, \(T _ { 1 } , T _ { 2 }\) and \(T _ { 3 }\).
In Figure 2, the capacity of each pipe is given by the number not circled on each edge. The numbers in circles represent an initial flow.

1. Add a supersource, supersink and appropriate weighted edges to Figure 2. (2 marks)
2. 1. Use the initial flow and the labelling procedure on Figure 3 to find the maximum flow through the network. You should indicate any flow augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
   2. State the value of the maximum flow and, on Figure 4, illustrate a possible flow along each edge corresponding to this maximum flow.
3. Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-18_1077_1246_1475_395}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_1049_1264_308_386}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_835_1011_1738_171}
   \end{figure}
   \includegraphics[max width=\textwidth, alt={}]{5123be51-168e-4487-8cd3-33aee9e3b23f-19_688_524_1448_1302}
   \includegraphics[max width=\textwidth, alt={}]{5123be51-168e-4487-8cd3-33aee9e3b23f-20_2256_1707_221_153}