7.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-4_759_1529_715_267}
\end{figure}
Figure 1 shows a capacitated, directed network. The unbracketed number on each arc indicates the capacity of that arc, and the numbers in brackets show a feasible flow of value 68 through the network.
- Add a supersource and a supersink, and arcs of appropriate capacity, to Diagram 1 below.
\section*{Diagram 1}
\includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-4_684_1531_1816_267}
- Find the values of \(x\) and \(y\), explaining your method briefly.
- Find the value of cuts \(C _ { 1 }\) and \(C _ { 2 }\).
Starting with the given feasible flow of 68,
- use the labelling procedure on Diagram 2 to find a maximal flow through this network. List each flow-augmenting route you use, together with its flow.
\section*{Diagram 2}
\includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-5_647_1506_612_283}
- Show your maximal flow on Diagram 3 and state its value.
\section*{Diagram 3}
\includegraphics[max width=\textwidth, alt={}, center]{eabe577b-80d9-45f8-a8e8-0c3b139b96a8-5_654_1511_1567_278}
- Prove that your flow is maximal.