Open and Closed Walks in Graph Theory

Introduction

• Discusses the concepts of open walks and closed walks in a graph.
• Important for understanding differences in other graph concepts such as trails and paths.

Definitions

Walk

• A sequence of vertices and edges connecting them in a graph.
• Represented by listing vertices in the order they are visited (e.g., A, B, C, E).

Open Walk

• Starts and ends at different vertices.
• Example: A walk from A to B to C to E is open if it does not return to A.

Closed Walk

• Starts and ends at the same vertex.
• Example: A walk from A to B to C to E and back to A.

Properties of Walks

• Revisiting Vertices: A vertex can be visited multiple times.
  • Example: Starting at D, visiting E, then C, then returning to D multiple times is allowed.
• Revisiting Edges: An edge can be traversed multiple times in a walk.
  • Example: Walking from the same vertices along the same edge more than once.

Differences in Future Topics

• Paths and trails will introduce restrictions not present in walks.
• Walks are unrestricted and can include revisiting both vertices and edges.

Summary

• The nature of a walk (open or closed) is determined by the starting and ending vertices.
  • Closed Walk: First and last vertices are the same.
  • Open Walk: First and last vertices are different.

Understanding this foundational concept of walks in graphs will aid in comprehending more constrained movements like paths and trails.