Understand the idea.
Reachability, not just adjacency
A route follows existing edges between consecutive vertices. Count the edges traversed, rather than the written vertex labels: A–B–D has two edges. A simple path also avoids repeating vertices.
A network is connected if every pair of vertices can be joined by a route. They need not be directly adjacent. In a disconnected network, each maximal group of mutually reachable vertices is a component, including an isolated vertex on its own.
Testing a link failure
To find components, start at one vertex and repeatedly mark neighbours of already marked vertices. Continue until no new vertex can be reached, then repeat from an unmarked vertex to find the next component.
A bridge is an edge whose removal increases the number of components. Test whether its endpoints still have an alternative route after removal; if they do, their connection has not depended on that edge alone.
See the method in action.
Follow a specified route
Give a three-edge route from A to C through D.
Worked solution
- 1
Begin by passing through D.
AD is an edge in the supplied network.
- 2
Continue from D to B and C.
Both DB and BC exist, so each consecutive move is permitted.
- 3
Check the full route and length.
The route uses AD, DB and BC, visits D, and finishes at C.
A–D–B–C.
Find the components
Vertices A–F have edges AB, BC and DE. Find the components.
Worked solution
- 1
Follow the links from A.
AB and BC connect these three vertices, and no given edge leaves their group.
- 2
Find the next reachable group.
DE joins D and E but has no link to A, B or C.
- 3
Include the remaining isolated vertex.
F has no incident edge, so it forms a component by itself alongside the two groups already found.
{A,B,C}, {D,E}, {F}: 3 components.
Try it yourself.
Check your understanding
The only edges are AB, BC, CD. What happens if BC is removed?
Show answer & reasoning
Components {A,B} and {C,D}.
Removing BC leaves only AB and CD, so the components are and . There is no alternative route between the two groups, making BC a bridge in the original network.
Keep practising
6 worksheet questions · 14 marks
Read the questions online
- Give two routes from A to C.2 marks
- Is A–C–E a valid route?2 marks
- Name all components.2 marks
- A cycle has edges AB, BC, CD, DA. Does deleting AB disconnect it?2 marks
- Three components contain 3,4,2 vertices. What is the minimum number of links to connect all of them?3 marks
- Edges are AB, BC, CA, CD, DE. List the bridges.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Give two routes from A to C.2 marks
Check consecutive pairs against the edges.
For example A–B–C and A–D–E–C. - Is A–C–E a valid route?2 marks
AC is not an edge.
No. - Name all components.2 marks
The square is joined; E is isolated.
{A,B,C,D} and {E}. - A cycle has edges AB, BC, CD, DA. Does deleting AB disconnect it?2 marks
A–D–C–B remains an alternative route.
No. - Three components contain 3,4,2 vertices. What is the minimum number of links to connect all of them?3 marks
One link can merge two components.
A second joins the third.
2. - Edges are AB, BC, CA, CD, DE. List the bridges.3 marks
Triangle edges have alternatives.
CD and DE are the only links to their later sections.
CD and DE.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum
Vertices, edges and connections; drawing networks from situations; interpreting connectedness and routes; comparing representations; using networks to explain practical relationships.
Official curriculum source ↗