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Classify a route: walk, trail, path, circuit or cycle

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Routes use edges

A walk is a sequence of adjacent vertices, together with the edges used. A trail repeats no edge. A path repeats no vertex. A circuit is a closed trail; a cycle is a circuit with no repeated vertex apart from its start at the end. These classes overlap: every cycle is also a circuit, trail and walk. When asked for one class, give the most specific one

Year 11, VCE Units 1 & 2, Chapter 2, 4 difficulty levels, No calculatorBuild a worksheet

Route classes

Check the actual edges before classifying the sequence

  1. Walk and trail A walk follows adjacent edges. A trail is a walk without repeated edges
  2. Path and cycle A path has no repeated vertex. A cycle returns to its start without repeating another vertex
  3. Circuit A closed trail may revisit intermediate vertices, so it need not be a cycle
  4. Overlapping classes A cycle is also a circuit, trail and walk. Give the most specific class when one answer is requested

Worked example

Edges LM, MN, NL and LP exist. Classify L-M-N-L-P as specifically as possible

\[L-M-N-L-P\]
  1. \(\displaystyle LM,MN,NL,LP\)Every step is an existing edge, so the sequence is a walk
  2. \(\displaystyle 4\text{ distinct edges}\)No edge repeats, so it is a trail
  3. \(\displaystyle L\ne P\)The trail is open
  4. \(\displaystyle L\text{ repeats}\)It is not a path. The most specific class is an open trail that is not a path

Common mistakes

Reversing an edge
✕\(\displaystyle BC\ne CB\)✓\(\displaystyle \{B,C\}=\{C,B\}\)
In an undirected simple graph these names refer to the same edge

Difficulty levels

introDistinguish an open path from a trail with a repeated vertex
standardClassify open and closed routes
challengeCheck several routes, including a non-adjacent step
examSelect the route of the requested most specific class

Questions about this topic

Can two classifications both be true?+

Yes. A path is a trail and a walk. A cycle is a circuit, trail and walk. The practice requests the most specific class

Does selecting a route assess my own construction?+

No. Generated questions classify or recognise supplied routes. Independent construction and explanation are separate ungraded work

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 2: Graph theory

In this chapter

Count the vertices, edges and loops of a graphRepresent a network in a matrixIdentify the network a connection matrix describesDegrees, the degree sum, and even and odd vertices2B Graph families and structure: Graph types, subgraphs, connection models and method selectionConnected graphs and bridgesIdentify isomorphic graphsTrees and spanning treesGraph families and tree conditionsPlanar graphs and drawings in planar formFaces and Euler's formula2C Planarity and graph proofs: Graph types, subgraphs, connection models and method selectionClassify a route: walk, trail, path, circuit or cycleEulerian trails and circuits: existence and identificationHamiltonian paths and cycles