The capacities of the arcs are as follows:
Arc from S to A: Capacity = 10
Arc from S to B: Capacity = 15
Arc from A to B: Capacity = 12
Arc from A to C: Capacity = 8
Arc from B to D: Capacity = 14
Arc from C to D: Capacity = 10
Arc from C to T: Capacity = 5
Arc from D to T: Capacity = 15
A. Maximal flow amount: The maximal flow is the sum of all flow values leaving the source node (S) or entering the sink node (T). In this case, it will be the flow leaving S, which is the sum of flow on arcs (S to A) and (S to B).
Maximal flow amount = Flow(S to A) + Flow(S to B)
B. The cut gives the maximal flow amount: The cut that gives the maximal flow amount is the minimum cut that separates the source node (S) from the sink node (T). The capacity of the minimum cut will be equal to the maximal flow amount.
C. The flow amounts on each arc:
Flow(S to A) = 10
Flow(S to B) = 10
Flow(A to B) = 10 (Capacity reached, so this arc is at its maximum flow)
Flow(A to C) = 0
Flow(B to D) = 10
Flow(C to D) = 0
Flow(C to T) = 0
Flow(D to T) = 10
D. Surplus capacities: The surplus capacity on each arc is the difference between the arc capacity and the flow amount.
Surplus(S to A) = Capacity(S to A) - Flow(S to A) = 10 - 10 = 0
Surplus(S to B) = Capacity(S to B) - Flow(S to B) = 15 - 10 = 5
Surplus(A to B) = Capacity(A to B) - Flow(A to B) = 12 - 10 = 2
Surplus(A to C) = Capacity(A to C) - Flow(A to C) = 8 - 0 = 8
Surplus(B to D) = Capacity(B to D) - Flow(B to D) = 14 - 10 = 4
Surplus(C to D) = Capacity(C to D) - Flow(C to D) = 10 - 0 = 10
Surplus(C to T) = Capacity(C to T) - Flow(C to T) = 5 - 0 = 5
Surplus(D to T) = Capacity(D to T) - Flow(D to T) = 15 - 10 = 5
E. The flow amounts through the intermediate nodes:
Flow through A = Flow(S to A) + Flow(A to B) + Flow(A to C) = 10 + 10 + 0 = 20
Flow through B = Flow(S to B) + Flow(A to B) + Flow(B to D) = 10 + 10 + 10 = 30
Flow through C = Flow(A to C) + Flow(C to D) + Flow(C to T) = 0 + 0 + 0 = 0
Flow through D = Flow(B to D) + Flow(C to D) + Flow(D to T) = 10 + 0 + 10 = 20
Note: The above solution assumes that there are no other arcs or nodes connected to the given network.