June 2026
The 2025 VCE General Mathematics exams showed that Networks and Decision Mathematics was not a topic students could approach by memory alone.
Students needed to inspect graphs carefully, follow algorithms in the correct order, interpret paths and cycles accurately, identify bridges, calculate cuts and maximum flow, determine shortest paths, apply the Hungarian algorithm, and reason through project networks.
The strongest students were methodical.
They did not guess from the appearance of a diagram.
They did not count every edge near a cut.
They did not treat a Hamiltonian cycle as an Eulerian trail.
They did not reverse the order of the Hungarian algorithm.
They did not calculate float without considering earliest and latest start times.
They did not ignore dummy activities in precedence networks.
Networks reward careful inspection.
A small misread edge can change the whole answer.
Hamiltonian cycles required visiting every vertex once
Examination 1 Question 33 asked for the number of Hamiltonian cycles in a graph, starting from vertex E.
The report stated that only two Hamiltonian cycles could be created when starting from E:
E–D–C–B–A–F–E
E–F–A–B–C–D–E
These are the same cycle travelled in opposite directions, but because the question counted cycles starting from E, the two directions were represented as two possible cycles.
This question tested the definition of a Hamiltonian cycle.
A Hamiltonian cycle visits every vertex exactly once and returns to the starting vertex.
Students needed to trace possible routes without repeating vertices or missing any.
A graph may contain many paths, but far fewer Hamiltonian cycles.
Bridges required removal logic
Examination 1 Question 34 asked for the number of bridges in a graph.
The report defined a bridge as an edge whose removal disconnects the network. It stated that any of the top three edges, when removed, caused the graph to become disconnected. Therefore, there were three bridges.
This is a simple definition, but it requires exact testing.
A bridge is not just an edge that looks important. It is an edge that, if removed, separates the graph into disconnected parts.
Students should test bridges by asking:
If this edge disappears, can every vertex still be reached from every other vertex?
If not, the edge is a bridge.
This prevents students from counting edges based on visual prominence rather than graph structure.
Minimum spanning trees required selected edges only
Examination 1 Question 35 asked for the length of a minimum spanning tree that contained an edge of weight w.
The assessment guide gave the total:
5 + 6 + 4 + w + 2 + 4 + 5 + 3 + 6 + 5 + 4 = 44 + w
This question rewarded students who could construct or identify the minimum spanning tree, then add only the edges included in that tree.
A minimum spanning tree connects all vertices with the smallest possible total weight and contains no cycles.
The key is disciplined selection.
Students should not add every edge in the graph. They should not add edges simply because they are small if doing so creates a cycle. They must connect all vertices while avoiding unnecessary edges.
A minimum spanning tree is about connection with minimum total length.
Cut capacity required direction
Examination 1 Question 36 asked for the capacity of Cut 1 in a flow network.
The report stated that Cut 1 crossed six edges, but only four had the direction of flow from source to sink. Those four capacities were:
7, 3, 5 and 12
The capacity of the cut was:
7 + 3 + 5 + 12 = 27
This is a crucial maximum-flow concept.
Not every edge intersected by a cut is counted. The capacity of a cut is determined by the directed edges crossing from the source side to the sink side.
Students who simply add all edges crossing a cut may get the wrong answer.
Direction matters.
Maximum flow was linked to minimum cut
Examination 1 Question 37 asked for the maximum flow of water from source to sink.
The report identified the minimum cut as:
7 + 4 + 6 = 17
These were the three edges flowing directly into the sink.
Therefore, the maximum flow was 17 litres per minute.
This question depended on the max-flow min-cut idea: the maximum flow through a network is equal to the capacity of the minimum cut.
Students needed to inspect the network for the limiting set of edges.
A large capacity elsewhere in the network does not matter if a smaller bottleneck restricts flow into the sink.
Maximum flow questions are about bottlenecks.
Shortest path required disciplined route comparison
Examination 1 Question 38 asked for the shortest distance from home to school on a road network.
The report gave the shortest path as:
900 + 500 + 400 + 800 + 1100 = 3700
So the shortest distance was 3700 metres.
This kind of question can look easy, but students often lose marks by choosing a route that appears direct rather than calculating alternatives.
Shortest path is not necessarily the path with the fewest edges. It is the path with the smallest total weight.
Students should accumulate distances carefully and compare possible routes systematically.
The diagram must be read as weighted information, not just a picture.
The Hungarian algorithm required the correct order
Examination 1 Question 39 involved allocating four contractors to four sites at minimum cost using the Hungarian algorithm.
The question described two steps:
First, subtract the minimum entry in each row from each element in that row.
Then, using this new table, subtract the minimum entry in each column from each element in that column.
The report stated that the correct table was option B.
It also explained why the other options were wrong:
Option A was produced after row reduction only.
Option C was produced by column reduction only.
Option D was produced by doing column reduction first and then row reduction.
This question is one of the clearest examples of algorithm discipline.
The operations themselves were not difficult. The issue was following the stated order.
In decision mathematics, the order of algorithmic steps can determine the answer.
Algorithm questions tested process, not just final allocation
Question 39 did not ask students to complete the entire Hungarian algorithm and find the final assignment. It asked for the table after two specified steps.
This is important.
Students needed to stop at the required stage.
If a student knows the full algorithm but ignores the question’s exact request, they may answer something else.
Many General Mathematics algorithm questions test intermediate process.
What is the table after row reduction?
What is the table after column reduction?
Which zero assignments are possible?
What is the minimum number of lines needed?
What adjustment should be made next?
Students need to answer the stage asked for, not the stage they expected.
Precedence networks required dummy activities
Examination 1 Question 40 gave a precedence table for a 12-activity project and asked for the float time of Activity B.
The report stated that the network required a dummy activity between the end of B and the start of G. The latest start of Bwas 12, the earliest start was 4, and the float was therefore:
12 − 4 = 8
This question is important because it shows that precedence networks are not always drawn directly from a table without adjustment.
Dummy activities may be required to preserve precedence relationships correctly.
In this case, Activity G depended on B, D and F, while other activities had overlapping dependencies. The dummy activity helped represent the logic of the project accurately.
Students need to understand why dummy activities exist: they show dependency without adding time.
Float required earliest and latest times
The float of Activity B in Question 40 was 8 because:
latest start − earliest start = 12 − 4 = 8
Float is the amount of time an activity can be delayed without delaying the project completion time.
This is a project-management concept, not just a subtraction.
An activity with float is not on a critical path. A critical activity has zero float. If an activity has eight days of float, it can start up to eight days later than its earliest start without extending the project.
Students need to interpret the result in project terms.
Examination 2 networks included adjacency matrices
Examination 2 Question 16 involved a gym building with areas labelled E, R, W, C and S. Students were given an adjacency matrix with some entries labelled x, y and z.
The exam explained that the 2 in row E, column C meant that there were two ways of moving from the entry to the change room without passing through another area or backtracking.
This required students to match the network to the adjacency matrix.
An adjacency matrix entry records the number of direct connections between two vertices. If there are two direct ways between two areas, the entry is 2. If there is no direct connection, the entry is 0.
This is similar to matrices, but it sits naturally inside network reasoning.
Students need to track both the diagram and the matrix labels.
Eulerian ideas required vertex degrees
Examination 2 Question 17 involved a gym owner inspecting all walkways in a network and wanting to begin and end at station A.
Question 17a asked students to explain why the intended route must involve some repeated edges, with reference to vertex degrees.
For a route that starts and ends at the same vertex while using every edge exactly once, every vertex must have even degree. If the network contains vertices of odd degree, some edges must be repeated.
This is the Eulerian circuit idea.
The task was not simply to say that the route is impossible. It required reference to the degrees of the vertices.
High-scoring responses connected the practical route to the mathematical condition.
Route inspection required minimum repeated distance
Question 17b then asked for the minimum distance the gym owner would cover when completing the inspection.
This is a Chinese postman-style problem. If a closed route must traverse every edge but the network has odd-degree vertices, some edges must be repeated. The goal is to repeat the minimum additional distance needed to allow a closed inspection route.
Students needed to add the total edge length and the minimum repeated path required to pair odd vertices.
This kind of question rewards structure:
Find vertex degrees.
Identify odd vertices.
Find the shortest path needed to pair them.
Add repeated distance to the total network length.
The story is about inspecting walkways, but the mathematics is about making an Eulerian circuit possible.
Critical path networks required shared activities
Examination 2 Question 18 involved constructing a home gym. The activity network had 12 activities, A to L, and contained two critical paths.
Question 18a asked students to state the activities common to both critical paths.
This required students to identify both critical paths, then find their overlap.
A common mistake in critical path questions is to find one critical path and stop. But if the question says there are two critical paths, students need to consider both.
An activity common to both paths is especially important because delaying it will affect both critical sequences.
This is project logic, not just graph tracing.
Latest start time required backward reasoning
Question 18b asked for the latest start time for Activity E.
Latest start times are usually found through a backward pass from the project completion time. They identify the latest time an activity can begin without delaying the overall project.
This requires students to know the difference between earliest start and latest start.
Earliest start comes from working forward through prerequisites.
Latest start comes from working backward from project completion.
These are not interchangeable.
Students should label activity times clearly when working through critical path networks.
Longest float required comparing non-critical activities
Question 18c asked which activity had the longest float time.
This required students to calculate or infer float values across activities and identify the largest.
The activity with longest float has the greatest allowable delay before affecting project completion. It is usually well away from the critical path, but students should not guess. They need to calculate.
Float depends on the project network, not just the duration of the activity.
A short activity can have long float. A long activity can have no float if it lies on a critical path.
Crashing required reducing critical paths efficiently
Question 18d asked for the minimum additional cost to reduce the home gym construction project by three days.
The table listed activities that could be reduced, with maximum reduction times and additional cost per day:
A: 2 days, $500 per day
F: 4 days, $150 per day
G: 4 days, $150 per day
H: 2 days, $300 per day
K: 1 day, $100 per day
This type of question requires strategic crashing.
Students cannot simply choose the cheapest activity overall. They must reduce the duration of the project, which means reducing activities on the critical path or paths. If there are two critical paths, reducing only one path may not reduce the overall project time unless the other path is also addressed.
The minimum-cost solution depends on which activities affect the project completion time.
Crashing is cost optimisation under dependency constraints.
Critical path questions required project interpretation
Critical path questions in 2025 required students to interpret activity networks as projects.
An activity’s duration is not just a number. It contributes to path length. A predecessor relationship is not just an arrow. It controls when later activities can begin. A critical path is not merely the longest route; it determines the minimum project completion time.
Students should always ask:
Which activities must occur before this one?
Which path determines completion time?
Which activities have zero float?
If this activity is shortened, does the project actually finish earlier?
If there are multiple critical paths, are both affected?
This prevents mechanical errors.
Networks rewarded terminology accuracy
The 2025 questions used several network terms:
Hamiltonian cycle
bridge
minimum spanning tree
cut capacity
maximum flow
shortest path
Hungarian algorithm
dummy activity
float
adjacency matrix
Eulerian circuit condition
critical path
latest start time
crashing
These terms are not interchangeable.
A Hamiltonian cycle visits every vertex once.
An Eulerian trail or circuit is concerned with edges.
A bridge disconnects a graph when removed.
A minimum spanning tree connects all vertices without cycles.
A cut limits flow from source to sink.
A critical path determines project duration.
Students need to know what each term is asking them to inspect.
Why Networks errors happen
Networks errors often happen because students rely on visual impressions.
They count an edge as a bridge because it looks isolated.
They add the wrong edges across a cut.
They choose the path with fewer edges rather than lower total distance.
They follow the Hungarian algorithm in the wrong order.
They forget dummy activities.
They calculate float from the wrong start time.
They shorten a cheap activity that does not reduce the project duration.
They confuse Hamiltonian and Eulerian conditions.
These errors are preventable with method.
Networks are visual, but they are not guesswork.
What future General Mathematics students should learn from 2025
The 2025 VCE General Mathematics exams show that Networks and Decision Mathematics preparation needs to focus on process and interpretation.
Students should practise:
- tracing Hamiltonian cycles without repeating vertices
- identifying bridges by testing disconnection
- constructing minimum spanning trees without cycles
- calculating cut capacity using correct edge direction
- applying maximum-flow minimum-cut reasoning
- finding shortest paths by total weight, not appearance
- following the Hungarian algorithm in the stated order
- recognising intermediate algorithm stages
- drawing precedence networks from tables
- using dummy activities appropriately
- calculating earliest start, latest start and float
- interpreting adjacency matrices for networks
- using vertex degrees to justify repeated edges
- solving inspection-route problems by pairing odd vertices
- identifying multiple critical paths
- crashing projects by reducing critical path duration at minimum cost
These are not separate tricks.
They are forms of network reasoning.
How ATAR STAR approaches Networks and Decision Mathematics
At ATAR STAR, Networks and Decision Mathematics is taught as methodical graph analysis.
Students learn to inspect diagrams systematically, label vertex degrees, trace paths carefully, follow algorithms step by step, and interpret project networks in context. They practise maximum flow, shortest path, spanning trees, Hamiltonian and Eulerian reasoning, Hungarian algorithm problems and critical path crashing with a focus on avoiding common VCAA traps.
The 2025 Examination Reports confirm why this matters. High-scoring students did not simply recognise network terminology.
They followed the structure of the graph.
That is what VCE General Mathematics rewards.