A __________ __________ touches every vertex exactly once, but does not return to its origin.

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A. Hamiltonian circuit
B. Hamiltonian path
C. Euler circuit
D. Euler path

To determine the correct answer, we need to understand the definitions of various graph theory terms and their properties.

1. Hamiltonian Circuit: A Hamiltonian circuit is a path in a graph that visits every vertex exactly once and returns to its starting vertex. In other words, it forms a closed loop.

2. Hamiltonian Path: A Hamiltonian path is a path in a graph that visits every vertex exactly once, but does not return to its starting vertex. In other words, it does not form a closed loop.

3. Euler Circuit: An Euler circuit is a circuit that traverses every edge of a graph exactly once and returns to its starting vertex.

4. Euler Path: An Euler path is a path that traverses every edge of a graph exactly once, but does not return to its starting vertex.

In the given question, it is mentioned that the path does not return to its starting vertex. Therefore, the correct answer is:

B. Hamiltonian path