Unit 4

🕸️ Networks & Decision Mathematics

Graphs, trees, paths and scheduling problems. A high-value section of the exam, learn to read networks, find optimal routes, plan projects efficiently and assign workers to minimise cost.

7 subtopics

7 of 7 ready to study now 🎉

Appears every exam year

New to Networks? Start with Graph Theory Basics to learn the key vocabulary, then work through in order. Already know the foundations? Jump straight to the subtopic you need, each one stands alone.

Graph Foundations

Graph Theory Basics

Vertices, edges, degree and types of graphs. The vocabulary you need before anything else in this topic makes sense.

vertices & edges degree connected graphs trees: n−1 edges

Euler Paths & Circuits

Can you trace a graph without lifting your pen? Learn the odd/even vertex rule and Euler's formula: v − e + f = 2.

traversability odd vertices Euler's formula v − e + f = 2

Optimisation, Trees & Paths

Minimum Spanning Tree

Connect all nodes using the least total weight of edges. Uses Kruskal's algorithm, sort edges, add if no cycle forms, stop at n−1 edges.

Kruskal's algorithm weighted graphs n−1 edges cycles

Find the minimum-weight route between two points in a network. Uses Dijkstra's algorithm or the inspection method.

Dijkstra's algorithm inspection method box the minimum

Scheduling, Project Planning

Critical Path Analysis

Find the minimum time to complete a project. Use a forward pass to work out the Earliest Start Time of each task, take the MAX at any join.

EST forward pass critical path activity networks take the MAX

How long can a task be delayed without pushing the whole project out? Backward pass to find LST, then Float = LST − EST. Covered in the Critical Path page.

LST backward pass Float = LST − EST slack time non-critical tasks

Assignment Problems

Hungarian Algorithm

Assign workers to jobs (or machines to tasks) to minimise total cost or time. A set process: row reduce → column reduce → cover zeros with minimum lines → assign.

row reduction column reduction covering zeros optimal assignment minimisation 3×3 matrix

Test Yourself

Sparky's Workshop, Escape Room

6 challenges covering MST, Critical Path, Float, Hungarian Algorithm, Time Zones and Business Maths. Work through all checkpoints to escape.

Node Zero, Escape Room

The school network has crashed. Diagnose the topology, degree sums, tree edge counts and Euler's planar formula, to restore every server.

Euler's District, Escape Room

Deploy city inspection robots by verifying odd-degree counts, traversability rules and planar graph face counts across six districts.

GPS Down, Escape Room

Stranded in the outback with a dead GPS and a paper map. Calculate shortest routes using inspection to reach the campsite before dark.