Understand the meaning of source node, sink node, cut, minimum cut and maximum flow.

Which cut separates the source from the sink and has the minimum capacity in the water distribution network shown?

A distribution network transports goods from a warehouse (source S) to a retail centre (sink T) through intermediate depots. The diagram shows the network with arc capacities in units per hour. (a) Identify the cut that separates {S, A} from {B, C, T} and calculate its capacity. [2 marks] (b) Determine whether this is the minimum cut for the network. Justify your answer. [2 marks]

A mining company transports ore from the mine \( M \) (source) to the processing plant \( P \) (sink) through a network of conveyor belts. The network is shown below, with edge capacities in tonnes per hour. (a) Identify one cut in the network and calculate its capacity. (2 marks) (b) Use your answer from part (a) to determine whether the maximum flow from \( M \) to \( P \) could be 34 tonnes per hour. Justify your answer. (2 marks)

Which of the following sets of edges forms a cut separating the source \(S\) from the sink \(T\) in the flow network shown below?

A water distribution network supplies water from a treatment plant at source \(S\) to a reservoir at sink \(T\). The diagram shows the maximum flow capacity (in kilolitres per hour) along each pipe. (a) Calculate the capacity of cut \(C_1\) shown in the diagram. (2 marks) (b) Determine the maximum flow from \(S\) to \(T\). Show your working. (2 marks)