https://file.aiquickdraw.com/mathbot/file/1730179620774ylxz6d87.jpg

To construct the Hamiltonian circuit using the *nearest neighbor algorithm* as directed in the image, starting from the vertex on the far right, we would follow these steps:

### Steps Using the Nearest Neighbor Algorithm:
1. **Choose the Starting Vertex:** Begin at the vertex on the far right (vertex labeled with the number 10).
2. **Find the Nearest Neighbor:** At each step, move to the closest vertex that hasn’t been visited yet. If there are multiple options with the same distance, choose any of them.
3. **Continue Until All Vertices are Visited:** Keep selecting the nearest vertex that hasn’t been visited until all vertices are visited exactly once.
4. **Return to the Starting Point:** Once all vertices are visited, return to the starting point to complete the circuit.

### Example Path:
Following this algorithm step-by-step from vertex 10:
- Start at vertex **10**.
- Move to the nearest neighbor, say **12**.
- From **12**, the closest vertex could be **2**.
- Continue this process until all vertices are visited.
- Finally, return to **10** to complete the Hamiltonian circuit.

Would you like a detailed list of each step, or have any specific questions?

### Related Questions:
1. How would the circuit differ if you started from a different vertex?
2. What is the importance of the Hamiltonian circuit in graph theory?
3. Could there be multiple Hamiltonian circuits for this graph?
4. How does the nearest neighbor algorithm work, and why is it not always optimal?
5. What are the practical applications of Hamiltonian circuits?

### Tip:
For graphs with many vertices, algorithms like the *nearest neighbor* might not give the shortest path but still provide a quick solution to find a Hamiltonian circuit.

Draw the circuit produced using the nearest neighbor algorithm starting at the vertex on the far right.

Learn how to use the nearest neighbor algorithm to construct a Hamiltonian circuit, visiting each vertex exactly once and returning to the starting point. This step-by-step guide explains the graph theory concepts behind this algorithm, suitable for high school students.

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