Walks, Trails, Paths, Cycles and Circuits in Graph

Walks, trails, paths, cycles, and circuits in a graph are sequences of vertices and edges with different properties. Some allow repetition of vertices and edges, while others do not.

Walk

A walk in a graph is a sequence of vertices and edges where both edges and vertices can be repeated. The length of the walk refers to the number of edges covered in the sequence. A graph can contain multiple walks.

There are two key points to note about a walk:

1. Edges can be repeated.
2. Vertices can be repeated.

• Here, 1-> 2-> 3-> 4-> 2-> 1-> 3 is a walk.

Walk can be open or closed.

Types of Walks

There are two types of walks:

Open Walk: An open walk is a walk in which the starting and ending vertices are different. In other words, for a walk to be considered open, the origin and terminal vertices must not be the same. Additionally, the length of the walk must be greater than 0.

Closed Walk: A closed walk occurs when the starting and ending vertices are identical, meaning the walk starts and ends at the same vertex. For a walk to be classified as closed, the origin and terminal vertices must be the same. Similar to an open walk, the length of the walk must be greater than 0.

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. 

Trail

A trail is an open walk in which no edge is repeated, though vertices may be repeated.

There are two types of trails:

1. Open Trail: A trail is considered an open trail if the starting and ending vertices are different.
2. Closed Trail: A trail is a closed trail if the starting and ending vertices are the same.

In a trail, the key point to remember is that: Edges cannot be repeated, but vertices 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 

Circuit

A circuit can be described as a closed trail in graph theory, where no edge is repeated, but vertices can be repeated.

Thus, the key characteristics of a circuit are:

• Edges cannot be repeated.
• Vertices can be repeated.

In other words, a circuit is a closed traversal of a graph where each edge is used exactly once, but a vertex may appear more than once.

Here 1-> 2-> 4-> 3-> 6-> 8-> 3-> 1 is a circuit.
Circuit is a closed trail.

Path

A path is a trail in which neither vertices nor edges are repeated.

In other words, when traversing a graph along a path, each vertex and each edge is visited exactly once. Since a path is also a trail, it is inherently an open walk unless stated otherwise.

Another definition of a path is a walk with no repeated vertices. This automatically implies that no edges are repeated, making it unnecessary to explicitly mention edge repetition in the definition.

Key characteristics of a path:

• Vertices are not repeated.
• Edges are not repeated.

Here 6->8->3->1->2->4 is a Path 

Cycle

A cycle in graph is a closed path, meaning that it starts and ends at the same vertex while ensuring that no other vertices or edges are repeated.

In other words, a cycle is formed by traversing a graph such that: No vertex is repeated, except for the starting and ending vertex, which must be the same and No edge is repeated.

Key characteristics of a cycle:

• Edges cannot be repeated.
• Vertices cannot be repeated, except for the first and last vertex, which must be the same.

Here 1->2->4->3->1 is a cycle. 
Cycle is a closed path.

Table for Walk, Trail and Path

The table below represents the repetition of edges and vertices in walk, trail and path.

Solved Examples

Question 1. Find a trail from A to D for below graph

A - B

| |

C - D

One possible trail is: A → B → D.

Question 2. Find the shortest path from A to C.

A - B - C

| |

D - E

The shortest path is: A → B → C.

Question 3. Finding All Possible Paths

A - B - C

| |

D - E

The path is : A → B → C

Practice Problems

Graph for Reference:

A
/ | \
B C D
\ | /
E
|
F

1. Find a path from A to F.
2. Find a simple path from B to F.
3. Find a trail that uses every edge exactly once.
4. Identify a cycle in the graph.
5. Find the shortest path from A to E.
6. List all possible paths from E to F.
7. Find a trail from C to F.
8. Find the longest simple path in the graph.
9. Count the number of distinct cycles in the graph.
10. Find a path from A to F that consists of exactly 3 edges.