Topic summary · Decision maths (D2)
Flows in Networks summary
Network flows move material from source to sink under capacity constraints. Max-flow equals min-cut capacity.
Files: D2-Flows in Networks.pdf · D2-Flows in Networks answers.pdf · free public downloads · diagrams in the PDFs
Key learning points
- Edge flow ≤ capacity; conserve flow at intermediate vertices.
- Value of a flow equals net leaving the source.
- Max-flow min-cut theorem links flow value to cut capacity.
- Augmenting paths increase flow until none remain.
Worked example 1 — Min cut
A cut has capacity 12 and a flow of value 12 is found. Conclusion?
Working
By the max-flow min-cut theorem.
Worked example 2 — Edge flow ≤ capacity; conserve flow at intermediate
Use this method from the Flows in Networks sheet: Edge flow ≤ capacity; conserve flow at intermediate vertices.
Working
Each key learning point matches a question type on the printable worksheet.
Worked example 3 — Value of a flow equals net leaving the source
Use this method from the Flows in Networks sheet: Value of a flow equals net leaving the source.
Working
Each key learning point matches a question type on the printable worksheet.
Worked example 4 — Max-flow min-cut theorem links flow value to cut capaci
Use this method from the Flows in Networks sheet: Max-flow min-cut theorem links flow value to cut capacity.
Working
Each key learning point matches a question type on the printable worksheet.
Worked example 5 — Augmenting paths increase flow until none remain
Use this method from the Flows in Networks sheet: Augmenting paths increase flow until none remain.
Working
Each key learning point matches a question type on the printable worksheet.
Scaffolded Learning Sheet and answers (PDFs)
This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.