In graph theory, a path that visits every vertex exactly once is called:A. Eulerian pathB. Hamiltonian pathC. Simple pathD. Directed path

Step 1: Break Down the Problem

We need to determine which option correctly defines a path in graph theory that visits every vertex exactly once.

Step 2: Relevant Concepts

1. Eulerian Path: A path through a graph that visits every edge exactly once.
2. Hamiltonian Path: A path through a graph that visits every vertex exactly once.
3. Simple Path: A path that does not repeat any vertices or edges.
4. Directed Path: A path in a directed graph where edges have a direction.

Step 3: Analysis and Detail

• Option A: Eulerian path does not satisfy our requirement as it focuses on edges, not vertices.
• Option B: Hamiltonian path fulfills the requirement by visiting every vertex exactly once.
• Option C: Simple path describes the nature of not repeating but does not specify visiting all vertices.
• Option D: Directed path refers to the nature of graph traversal and does not focus on vertex visitation.

Step 4: Verify and Summarize

Upon analysis, the only term that fits the description given in the question is the Hamiltonian path, as it explicitly states visiting every vertex once.

Final Answer

B. Hamiltonian path