1.
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\caption{Figure 1}
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Figure 1 shows a capacitated, directed network of pipes. The numbers in circles represent an initial flow from S to T . The other number on each arc represents the capacity, in litres per second, of the corresponding pipe.
- State the value of \(x\)
- State the value of \(y\)
- State the value of the initial flow.
- State the capacity of cut \(C _ { 1 }\)
- Find, by inspection, a flow-augmenting route to increase the flow by four units. You must state your route.
The flow-augmenting route from (d) is used to increase the flow from S to T .
- Prove that the flow is now maximal.
A vertex restriction is now applied so that no more than 12 litres per second can flow through E.
- Complete Diagram 1 in the answer book to show this restriction.
- State the value of the maximum flow through the network with this restriction.