Mathematics | Walks, Trails, Paths, Cycles and Circuits in Graph

Prerequisite – Graph Theory Basics – Set 1
1. Walk –
A walk is a sequence of vertices and edges of a graph i.e. if we traverse a graph then we get a walk.
Vertex can be repeated
Edges can be repeated

Here 1->2->3->4->2->1->3 is a walk

Walk can be open or closed.
Walk can repeat anything (edges or vertices).

Open walk-A walk is said to be an open walk if the starting and ending vertices are different i.e. the origin vertex and terminal vertex are different.
Closed walk-A walk is said to be a closed walk if the starting and ending vertices are identical i.e. if a walk starts and ends at the same vertex, then it is said to be a closed walk.
In the above diagram:
1->2->3->4->5->3-> is an open walk.
1->2->3->4->5->3->1-> is a closed walk.

2. Trail –
Trail is an open walk in which no edge is repeated.
Vertex can be repeated

Here 1->3->8->6->3->2 is trail
Also 1->3->8->6->3->2->1 will be a closed trail

3. Circuit –
Traversing a graph such that not an edge is repeated but vertex can be repeated and it is closed also i.e. it is a closed trail.
Vertex can be repeated
Edge not repeated

Here 1->2->4->3->6->8->3->1 is a circuit

Circuit is a closed trail. 
These can have repeated vertices only.

4. Path –
It is a trail in which neither vertices nor edges are repeated i.e. if we traverse a graph such that we do not repeat a vertex and nor we repeat an edge. As path is also a trail, thus it is also an open walk.
Vertex not repeated
Edge not repeated

Here 6->8->3->1->2->4 is a Path

5. Cycle –
Traversing a graph such that we do not repeat a vertex nor we repeat a edge but the starting and ending vertex must be same i.e. we can repeat starting and ending vertex only then we get a cycle.
Vertex not repeated
Edge not repeated

Here 1->2->4->3->1 is a cycle.

Cycle is a closed path. 
These can not have repeat anything (neither edges nor vertices).

Note that for closed sequences start and end vertices are the only ones that can repeat.

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