QCE General Mathematics - Unit 4 - Networks and decision mathematics 2
Flow Networks | QCE General Mathematics
Learn source nodes, sink nodes, cuts, minimum cuts and maximum flow for QCE General Mathematics flow networks.
Updated 2026-05-18 - 5 min read
QCAA official coverage - General Mathematics 2025 v1.3
Exact syllabus points covered
- Understand the meaning of source node, sink node, cut, minimum cut and maximum flow.
- Use a flow network diagram to identify a cut.
- Determine the capacity of a cut.
- Solve small-scale practical problems involving flow networks (up to 8 possible cuts), including determining the minimum cut and the maximum flow.
A flow network models movement from a source to a sink through directed edges with capacities. It can represent water pipes, traffic, internet data, goods or passengers.
Original Sylligence diagram for general flow network.
Source, sink and capacity
The source is where flow begins. The sink is where flow ends. Each directed edge has a capacity showing the maximum amount that can travel along that edge.
Cuts
A cut separates the source from the sink. The capacity of a cut is the total capacity of edges crossing from the source side to the sink side.
This is the maximum-flow-minimum-cut idea. In General Mathematics, flow-network questions are small enough to solve by listing possible cuts and comparing capacities.
Worked example
Common traps
Path flow and bottlenecks
One way to estimate flow is to look at possible source-to-sink paths. The maximum that can be sent along a path is limited by the smallest capacity on that path. This smallest capacity is the bottleneck.
However, path-by-path reasoning can become messy because different paths may share edges. Cut analysis is usually cleaner for small QCE questions.
Cut capacity details
A cut divides the vertices into two groups: one containing the source and one containing the sink. Only count capacities of edges crossing from the source side to the sink side. Ignore edges crossing backwards.
| Cut edge direction | Count it? | |---|---| | source side to sink side | yes | | sink side to source side | no | | stays within one side | no |
Maximum-flow-minimum-cut theorem
If the minimum cut has capacity $18$, then no flow greater than $18$ can pass through the network. If a flow of $18$ can also be found, then $18$ is the maximum flow.
Depth: capacity and flow
A flow network has directed edges with capacities. The capacity is the maximum amount that can travel along that edge in the given direction. Flow must obey two rules:
- The flow on an edge cannot exceed its capacity.
- At intermediate vertices, total inflow equals total outflow.
The source produces flow and the sink receives flow.
Path bottlenecks
The capacity of a path is limited by its smallest edge capacity. This is called the bottleneck.
Finding a maximum flow often involves identifying several paths and adding flows without exceeding edge capacities.
Cuts
A cut separates the source from the sink. The capacity of a cut is the total capacity of edges crossing from the source side to the sink side.
The maximum-flow minimum-cut theorem says that the value of the maximum flow equals the capacity of the minimum cut. In General Mathematics, this is often used to justify that a proposed flow is maximal.
Residual thinking
After sending flow along a path, unused capacity remains on each edge. For example, if an edge has capacity $10$ and currently carries $6$, it has residual capacity $4$. Some methods also allow reversing flow along used edges when improving a solution, but many school questions can be solved by careful inspection.
Reporting a maximum-flow answer
A complete answer should include:
- the value of the total flow out of the source
- the value of the total flow into the sink
- confirmation that intermediate vertices conserve flow
- a cut or bottleneck argument if asked to show the flow is maximum
Depth: conservation check
Flow conservation is the easiest way to find errors in a proposed flow. At every intermediate vertex:
$ \text{inflow}=\text{outflow} $
The source and sink are exceptions. The source has net outflow and the sink has net inflow.
Capacities belong to directed edges. If a diagram has arrows in both directions between two vertices, treat them as separate directed edges, each with its own capacity.