Trail

Definition

A trail in graph theory is a sequence of vertices connected by edges, where no edge is repeated. It is used to describe a path through a graph that follows specific rules.

5 Must Know Facts For Your Next Test

1. A trail can visit the same vertex more than once but cannot traverse the same edge more than once.
2. Every path is a trail, but not every trail is a path.
3. An Eulerian trail visits every edge of a graph exactly once.
4. If all edges in the graph are distinct, then a trail becomes a simple path.
5. To determine if a trail exists between two vertices, one needs to check for the presence of edges without repetition.

Review Questions

• What distinguishes a trail from other types of paths in graph theory?
• Can an Eulerian trail visit the same vertex more than once? Explain why or why not.
• How would you verify if a given sequence of vertices and edges forms a valid trail?

Related terms

Path:

A sequence of vertices where each adjacent pair is connected by an edge and no vertex is repeated.

Circuit:

A closed trail where the starting vertex and ending vertex are the same.

Eulerian Trail: A trail that visits every edge of the graph exactly once without repetition.