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

2. Network Flows – Restricted vertices
To deal with this, we need to redraw the network, with B replaced by new vertices B1 and B2. The arc B1B2 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 B1.
In the original network arcs go out of B into vertices C and D, so these arcs must come out of the new vertex B2. 4 A C 7 8 3 5 T S 8 10 9 B1 B2 D 10 8

3. Network Flows – Restricted vertices
The labelling procedure can now be applied to this new network in the usual way.
Draw in the excess capacities and potential backflows, with an initial flow of zero. 8 4 9 7 10 5 3 8 4 9 7 10 5 3 A C T S B1 B2 D

4. Network Flows – Restricted vertices
To deal with this, we need to redraw the network, with B replaced by new vertices B1 and B2. The arc B1B2 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 B1.
In the original network arcs go out of B into vertices C and D, so these arcs must come out of the new vertex B2. 4 A C 7 8 3 5 T S 8 10 9 B D 8 10

5. Network Flows – Restricted vertices
Now choose a flow augmenting path.
One possible path is SACT.
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. 8 4 9 7 10 5 3 A C 4 4 3 4 4 T S B1 B2 D

6. Network Flows – Restricted vertices
Now choose a flow augmenting path.
One possible path is SACT.
Arc AC is now saturated. 8 4 9 7 10 5 3 A C 4 4 3 4 4 T S B1 B2 D

7. Network Flows – Restricted vertices
Now choose another flow augmenting path.
One possible path is SB1B2DT.
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. 8 4 9 7 10 5 3 A C 4 4 3 4 4 T S 8 8 2 8 8 1 B1 B2 D 2

8. Network Flows – Restricted vertices
Now choose another flow augmenting path.
One possible path is SB1B2DT.
Arc B2D is now saturated. 8 4 9 7 10 5 3 A C 4 4 3 4 4 T S 8 8 2 8 8 1 B1 B2 D 2

9. Network Flows – Restricted vertices
Now choose another flow augmenting path.
One possible path is SAB1B2CT.
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. 8 4 9 7 10 5 3 A C 2 4 4 3 1 6 4 4 6 T S 2 3 1 2 8 8 2 8 10 8 1 B1 B2 D 2

10. Network Flows – Restricted vertices
Now choose another flow augmenting path.
One possible path is SAB1B2CT.
Arc B1B2 is now saturated. 8 4 9 7 10 5 3 A C 2 4 4 3 1 6 4 4 6 T S 2 3 1 2 8 8 2 8 10 8 1 B1 B2 D 2

11. Network Flows – Restricted vertices
Since arc AC and arc B1B2 are both now saturated, S and T are disconnected and so we have the maximum flow.
We can write the final flows on the diagram.
The total flow is 14. 8 4 9 7 10 5 3 4 6 A C 2 4 4 3 1 6 4 6 4 6 T S 2 3 1 2 2 8 8 2 2 8 10 8 1 8 8 B1 B2 D 2 8 10

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