Mathematics Standard • Year 11 • Networks • Lesson 3
Paths, Walks, Trails & Circuits · Master
Combine route classification with edge-use reasoning and identify why a candidate fails.
Practice
Unless a question says otherwise, treat each network as a simple undirected graph: edges are unordered vertex pairs, with no loops or duplicate edges. Show your reasoning in the answer space; an edge list is enough.
Q1 · Inspect the route. A network has edges AB, BC, CD, DA, AC, CE. Classify A–B–C–A–D–C. Does it repeat a vertex or edge? 3 marks
Q2 · Find a circuit. For that network, give a circuit of length 3 and list its edges. 2 marks
Q3 · Circuit versus Eulerian circuit. A route starts and finishes at the same vertex but repeats an edge. Can it be a circuit? What can it be called? 2 marks
Q1 · Inspect the route
A walk and trail of length 5: edges AB, BC, CA, AD, DC are distinct. Vertices A and C repeat, so it is not a path. It is not closed.
Q2 · Find a circuit
A–B–C–A is a circuit, using AB, BC and CA.
Q3 · Circuit versus Eulerian circuit
No. A circuit is a closed trail, so it cannot repeat an edge. It also cannot be an Eulerian circuit, which must use every edge of the network exactly once.