1. 8 4 9 7 10 8 8 5 3 The diagram below shows water flowing through a pipework system. The values on the edges are the capacities of water that they can carry. The diagram below shows water flowing through a pipework system. The values on the edges are the capacities of water that they can carry. S Network Flows – Restricted vertices Vertex B has a restricted capacity. A maximum of 10 units can flow through this vertex. 10 AC D T B

2. 8 4 9 7 10 8 8 5 3 S Network Flows – Restricted vertices 10 AC D T B1B1B1B1 B2B2B2B2 To deal with this, we need to redraw the network, with B replaced by new vertices B 1 and B 2. The arc B 1 B 2 has a capacity of 10. In the original network arcs from vertices S and A go into B, so these arcs must go into the new vertex B 1. In the original network arcs go out of B into vertices C and D, so these arcs must come out of the new vertex B 2.

3. The labelling procedure can now be applied to this new network in the usual way. S Network Flows – Restricted vertices 849 7 10 8 8 5 3 10 AC D T B1B1B1B1 B2B2B2B2 Draw in the excess capacities and potential backflows, with an initial flow of zero. 849 7 10 8 8 5 3 10 0 0 0 0 0 0 0 0 0 0

4. 8 4 9 7 10 8 8 5 3 S To deal with this, we need to redraw the network, with B replaced by new vertices B 1 and B 2. The arc B 1 B 2 has a capacity of 10. Network Flows – Restricted vertices 10 AC D T B In the original network arcs from vertices S and A go into B, so these arcs must go into the new vertex B 1. In the original network arcs go out of B into vertices C and D, so these arcs must come out of the new vertex B 2.

5. Now choose a flow augmenting path. One possible path is SACT. Now choose a flow augmenting path. One possible path is SACT. S Network Flows – Restricted vertices AC D T B1B1B1B1 B2B2B2B2 849 7 10 8 8 5 3 10 0 0 0 0 0 0 0 0 0 0 The minimum arrow along this path is 4, and so this the increase in the flow. We decrease all the arrows in the direction of this path by 4, and increase all the arrows against the direction of this path by 4. The minimum arrow along this path is 4, and so this the increase in the flow. We decrease all the arrows in the direction of this path by 4, and increase all the arrows against the direction of this path by 4. 4 4 0 4 3 4

6. Now choose a flow augmenting path. One possible path is SACT. Now choose a flow augmenting path. One possible path is SACT. S Network Flows – Restricted vertices AC D T B1B1B1B1 B2B2B2B2 849 7 10 8 8 5 3 10 0 0 0 0 0 0 0 0 0 0 Arc AC is now saturated. 4 4 0 4 3 4

7. Now choose another flow augmenting path. One possible path is SB 1 B 2 DT. Now choose another flow augmenting path. One possible path is SB 1 B 2 DT. S Network Flows – Restricted vertices AC D T B1B1B1B1 B2B2B2B2 849 7 10 8 8 5 3 10 0 0 0 0 0 0 0 0 0 0 The minimum arrow along this path is 8, and so this the increase in the flow. We decrease all the arrows in the direction of this path by 8, and increase all the arrows against the direction of this path by 8. The minimum arrow along this path is 8, and so this the increase in the flow. We decrease all the arrows in the direction of this path by 8, and increase all the arrows against the direction of this path by 8. 4 4 0 4 3 4 1 8 2 8 0 8 2 8

8. Now choose another flow augmenting path. One possible path is SB 1 B 2 DT. Now choose another flow augmenting path. One possible path is SB 1 B 2 DT. S Network Flows – Restricted vertices AC D T B1B1B1B1 B2B2B2B2 849 7 10 8 8 5 3 10 0 0 0 0 0 0 0 0 0 0 Arc B 2 D is now saturated. 4 4 0 4 3 4 1 8 2 8 0 8 2 8

9. Now choose another flow augmenting path. One possible path is SAB 1 B 2 CT. Now choose another flow augmenting path. One possible path is SAB 1 B 2 CT. S Network Flows – Restricted vertices AC D T B1B1B1B1 B2B2B2B2 849 7 10 8 8 5 3 10 0 0 0 0 0 0 0 0 0 0 The minimum arrow along this path is 2, and so this the increase in the flow. We decrease all the arrows in the direction of this path by 2, and increase all the arrows against the direction of this path by 2. The minimum arrow along this path is 2, and so this the increase in the flow. We decrease all the arrows in the direction of this path by 2, and increase all the arrows against the direction of this path by 2. 4 4 0 4 3 4 1 8 2 8 0 8 2 8 2 6 1 2 0 10 3 2 1 6

10. Now choose another flow augmenting path. One possible path is SAB 1 B 2 CT. Now choose another flow augmenting path. One possible path is SAB 1 B 2 CT. S Network Flows – Restricted vertices AC D T B1B1B1B1 B2B2B2B2 849 7 10 8 8 5 3 10 0 0 0 0 0 0 0 0 0 0 Arc B 1 B 2 is now saturated. 4 4 0 4 3 4 1 8 2 8 0 8 2 8 2 6 1 2 0 10 3 2 1 6

11. Since arc AC and arc B 1 B 2 are both now saturated, S and T are disconnected and so we have the maximum flow. S Network Flows – Restricted vertices AC D T B1B1B1B1 B2B2B2B2 849 7 10 8 8 5 3 10 0 0 0 0 0 0 0 0 0 0 We can write the final flows on the diagram. The total flow is 14. We can write the final flows on the diagram. The total flow is 14. 4 4 0 4 3 4 1 8 2 8 0 8 2 8 2 6 1 2 0 10 3 2 1 6 6 8 2 4 10 2 8 0 8 6

12. 8 4 9 7 10 8 8 5 3 We can now go back to the original diagram, and write on the final flows. S Network Flows – Restricted vertices 10 AC D T B 6 8 2 4 2 8 0 8 6