Mathematics Standard • Year 11 • Networks • Lesson 3
Paths, Walks, Trails & Circuits · Build
Distinguish repeated vertices from repeated edges and classify a route precisely.
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 · Key definitions. Complete: a trail has no repeated ______; a path has no repeated ______ or ______ (the start may equal the end for a closed path). 3 marks
Q2 · Classify a closed route. Use E={AB,BC,CA,AD}. Classify A–B–C–A as a walk, trail, path and/or circuit. 2 marks
Q3 · Count the length. For the same graph, classify A–B–A and give its length. 2 marks
Q1 · Key definitions
A trail has no repeated edges; a path has no repeated edges or vertices, except that a closed path starts and ends at the same vertex.
Q2 · Classify a closed route
It is a closed trail and circuit: it starts and ends at A and repeats no edge. It is also a closed path (a cycle), since the only repeated vertex is the start/finish A.
Q3 · Count the length
A closed walk of length 2. It repeats edge AB, so it is not a trail or circuit; the repeated internal vertex also means it is not a path.