The idea, step by step
Reachability, not just adjacency
A route follows edges between consecutive vertices. A–B–D has three vertex labels but uses two edges. A simple path has no repeated vertex. Always check that each consecutive pair is actually joined.
A network is connected when every pair of vertices can be joined by some route. Direct links between every pair are not needed. The separate reachable groups in a disconnected network are called components.
Testing a link failure
Start at a vertex and mark every vertex you can reach, continuing from newly marked vertices. If any remain unmarked, start a new component. This systematic procedure avoids mistaking a drawing gap for disconnection.
Removing an edge may or may not disconnect the network. If an alternative route still connects its endpoints, the group can stay joined. An edge whose removal increases the component count is called a bridge.
A common trap
Calling two non-adjacent vertices disconnected without checking routes.
Instead: An indirect route may connect them. Connectedness is about reachability.
See the thinking
Follow a specified route
Give a three-edge route from A to C through D.
- AD exists.
- DB and BC exist.
- The sequence uses three edges.
Find the components
Vertices A–F have edges AB, BC and DE. Find the components.
- A, B, C are linked.
- D, E are linked.
- F is isolated.
Check your understanding
The only edges are AB, BC, CD. What happens if BC is removed?
Show answer
Components {A,B} and {C,D}.
There is no other route linking the two sides.
Your turn
6 questions · 14 marks · Work at your own pace.
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 ↗