Understand the idea.
Arrows and weights have different jobs
A vertex represents a place or item; an edge records a connection. In a directed network, an arrow permits travel only in its shown direction. A separate reverse arrow is needed to permit direct travel back.
A weight may describe distance, time, cost or capacity. Read its units rather than measuring the drawing. An edge labelled twelve passengers per minute describes throughput; a long line on the page need not have a large weight.
Validate before adding
Before calculating a route total, check every consecutive connection and its direction. One missing or reversed arrow invalidates the route even when its proposed weights add correctly. A missing edge does not supply a free connection.
For a valid time or distance route, add the weights of the edges actually used. Record reverse directions separately if both are available; their weights need not match. The chosen interpretation must remain consistent throughout the route.
See the method in action.
A permitted journey
Find the time for A→B→C→D in the displayed network, whose weights are minutes.
Worked solution
- 1
Check every required arrow.
The displayed network supplies A to B, B to C and C to D in the required directions.
- 2
Read the three travel times.
The weights are minutes for the actual edges in this route.
- 3
Add the route weights.
Sequential travel times add along a permitted journey.
9 minutes.
A reverse edge is not supplied
Can a vehicle travel directly from C to B in this network?
Worked solution
- 1
Identify the required direct connection.
Direct travel from C to B requires an arrow whose start is C and finish is B.
- 2
Compare with the diagram.
The shown connection points from B to C; there is no reverse C-to-B edge.
- 3
Conclude about the journey.
The requested direct journey is not permitted because a directed edge cannot be traversed backwards.
No direct C→B journey is permitted.
Try it yourself.
Check your understanding
A connection is labelled 12 passengers/min. Does 12 represent distance?
Show answer & reasoning
No; it is a capacity/rate.
The units are passengers per minute, so twelve is a capacity or rate. A distance weight would use length units instead.
Keep practising
6 worksheet questions · 12 marks
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 ↗