Lecture on Networks: Trails and Paths

Introduction

• Discussion about different types of walks in a network.
• Focus on trails and paths, which are specific types of walks.

Walks

• Defined as any trip through a network.

Trails

• Definition: A trail is a walk where no road (edge) gets walked on more than once.
• Example: Starting at Point E, moving to A, B, C, E, and D without repeating any road.
• If a road is repeated, it is no longer a trail.
• Types:
  • Open Trail: Starts and finishes at different vertices.
  • Closed Trail: Starts and finishes at the same vertex, with no repeated edges.

Paths

• Definition: A path is a walk where no vertex is visited more than once.
• Example: Starting at C, going to B, E, and D without repeating any vertex.
• Types:
  • Open Path: Starts and finishes at different vertices.
  • Closed Path: Starts and finishes at the same vertex, but may have a repeated vertex as the first and last.
• Rule Adjustment: No repeated vertices except for the first and last in a closed path.

Examples and Classification

• Example walk: C -> D -> E -> C -> B
  • Open Walk: Does not start and finish at the same vertex.
  • Path Check: Repeated vertex (C) is not first and last, so not a path.
  • Trail Check: No repeated edges, so it is a trail.

Summary

• Importance of classifying walks into trails, paths, or neither based on provided criteria.
• Ability to identify and categorize given walks into these classifications.