The idea, step by step
Arrows and weights have different jobs
A vertex represents a location or item; an edge represents a connection. In a directed network the arrow permits movement in one direction only unless another reverse edge is given. A weight can describe time, cost, distance or capacity. Record its units.
A→B does not imply B→A. A missing edge does not mean a zero-cost journey; it means that direct connection is unavailable. A curved or long-looking line is still interpreted by its stated weight, not its drawn length.
Validate before adding
A route is a sequence of vertices. Check each consecutive pair against the arrows before adding weights. If one edge is absent or points the wrong way, the route is invalid even when the arithmetic is correct.
When writing an edge table, record each directed connection separately. Opposite directions can have different travel times. Include closures, permitted transport types or other restrictions that affect the model.
A common trap
The drawn longer line must take longer.
Instead: Use its labelled weight; the drawing is schematic.
See the thinking
A permitted journey
Find the time for A→B→C→D in the displayed network, whose weights are minutes.
- All three required arrows are present.
- Add .
A reverse edge is not supplied
Can a vehicle travel directly from C to B in this network?
- The diagram has B→C.
- It does not have C→B.
Check your understanding
A connection is labelled 12 passengers/min. Does 12 represent distance?
Show answer
No; it is a capacity/rate.
Read the units to identify the weight meaning.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- A→B takes 5 min, B→C takes 4 min, A→C takes 12 min. Find the A→B→C journey time.2 marks
- The only edges are A→B, B→C and A→C. Is C→B→A permitted?2 marks
- P→Q has capacity 8 vehicles/min. Interpret this label.2 marks
- An undirected edge X–Y weighs 7 km. Find the length X→Y→X when repetition is permitted.2 marks
- A→B costs 4 dollars while B→A costs 6 dollars. Explain why one undirected weight is unsuitable.2 marks
- Why must a missing connection not be recorded as a zero-weight available edge?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A→B takes 5 min, B→C takes 4 min, A→C takes 12 min. Find the A→B→C journey time.2 marks
Both arrows exist.
.
9 min. - The only edges are A→B, B→C and A→C. Is C→B→A permitted?2 marks
C→B is absent.
B→A is also absent.
No. - P→Q has capacity 8 vehicles/min. Interpret this label.2 marks
This is a capacity.
It does not state the travel time.
At most 8 vehicles per minute can use the directed connection P→Q. - An undirected edge X–Y weighs 7 km. Find the length X→Y→X when repetition is permitted.2 marks
An undirected edge can be traversed both ways.
.
14 km. - A→B costs 4 dollars while B→A costs 6 dollars. Explain why one undirected weight is unsuitable.2 marks
The model needs two directed edges.
One undirected value would lose information.
The cost depends on direction. - Why must a missing connection not be recorded as a zero-weight available edge?2 marks
Absent and available at zero cost are different.
It could produce an invalid optimum.
That would falsely create a free direct connection.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: matrices and networks. Original learning resources follow the approved Ascent Library chapter map.
Official curriculum source ↗