Walks, Trails, Paths, Circuits, and Cycles in Graph Theory: Exploring Graph Traversal

Learn about different types of graph traversals: walks, trails, paths, circuits, and cycles. This guide defines each term, explains their characteristics, and provides examples to illustrate the distinctions between these fundamental graph concepts.

Walks, Trails, Paths, Circuits, and Cycles in Graph Theory

Walks

A walk is a sequence of vertices and edges in a graph. You can think of it as a route through the graph. In a walk, you're allowed to repeat vertices and edges.

The length of a walk is the number of edges it contains.

Examples of Walks

(An illustrative graph should be included here, followed by three examples of walks in that graph, clearly indicating the sequence of vertices and the length of each walk.)

Types of Walks

  • Open Walk: The starting and ending vertices are different.
  • Closed Walk: The starting and ending vertices are the same (also called a circuit).

(A note that a walk of length 0 is called a trivial walk, and that open and closed walks can contain repeated edges and vertices should be added here.)

(An illustrative graph with examples of open and closed walks should be added here.)

Trails

A trail is an open walk where no edge is repeated. Vertices can be repeated.

(An illustrative graph with examples of open and closed trails should be included here.)

Circuits

A circuit is a closed walk where no edge is repeated. Vertices can be repeated. A closed trail is a circuit.

(An illustrative graph with an example of a circuit should be included here.)

Cycles

A cycle is a closed walk where neither edges nor vertices are repeated (except that the starting and ending vertices are the same).

(An illustrative graph with an example of a cycle should be included here.)

Paths

A path is an open walk where neither edges nor vertices are repeated.

(An illustrative graph with an example of a path should be included here.)

Relationships Between Walk Types

  • Every path is a trail, but not every trail is a path.
  • Every cycle is a circuit, but not every circuit is a cycle.

Directed Graphs

In a directed graph, the edges have a direction. For directed graphs, we use terms like "directed walk," "directed trail," "directed path," "directed circuit," and "directed cycle".

Example 1: Identifying Walk Types

(This example, identifying walks, trails, paths, circuits, and cycles in a given graph, is provided in the original text and should be included here. The solution explaining the classification of each sequence should be given.)

Example 2: Identifying Walk Types in a Different Graph

(This example, identifying walks, trails, paths, circuits, and cycles in a different given graph, is provided in the original text and should be included here. The solution explaining the classification of each sequence should be given.)

Example 3: Analyzing Directed Walks

(This example, analyzing sequences in a directed graph to determine directed walks, directed paths, and directed cycles, is given in the original text and should be included here. The solutions to all parts of the problem should be clearly stated.)

Conclusion

The concepts of walks, trails, paths, circuits, and cycles are fundamental in graph theory, providing a framework for describing and analyzing routes and connections within a graph.

Analyzing Walks in Directed Graphs

Identifying Directed Walks

(An illustrative directed graph should be included here.)

Let's determine which of these sequences represent valid directed walks in the graph above:

  1. D, B, E, C, B
  2. D, A, B, E, D
  3. D, A, B, E, D, B, A, D
  4. D, B, E, C, B, A, D

Solution:

  • Sequences 1 and 2 are directed walks because a directed walk can have repeated vertices and edges.
  • Sequence 3 is not a directed walk because there's no edge from A to B.
  • Sequence 4 is not a directed walk because there's no edge from B to A.

Determining Lengths of Directed Walks

The length of a directed walk is the number of edges in the walk. Since only sequences 1 and 2 are directed walks, their length will be calculated.

The length of both directed walks 1 and 2 is 4.

Identifying Directed Paths

A directed path is a directed walk with no repeated vertices. Let's check if sequences 1 and 2 are directed paths.

Neither sequence 1 nor sequence 2 is a directed path because they both contain repeated vertices (sequence 1 repeats B, sequence 2 repeats D).

Identifying Directed Cycles

A directed cycle is a closed directed walk with no repeated vertices (except the start and end vertex, which are the same). Let's see if sequences 1 and 2 form directed cycles.

Neither sequence 1 nor sequence 2 is a directed cycle because they both contain repeated vertices (sequence 1 repeats B, and sequence 2 repeats D).

Conclusion

This example illustrates how to identify different types of walks (walks, paths, and cycles) within a directed graph, highlighting the importance of considering edge direction and vertex repetition.