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A **Hamilton path** is a path in a graph that visits each vertex exactly once. Let's analyze the graph and the given options to find two different Hamilton paths.

**Vertices in the graph**:  
- \( A \), \( B \), \( C \), \( D \), \( E \), \( F \), and \( G \)

We need to verify which of the given options are Hamilton paths, ensuring that they visit all vertices exactly once without repetition.

Let's check each option:

1. **Option A: \( B, F, C, A, D, B, E \)**  
   - This option revisits vertex \( B \) twice, so it is **not a Hamilton path**.

2. **Option B: \( C, B, F, A, D, G, E \)**  
   - This path visits all vertices exactly once, making it a **valid Hamilton path**.

3. **Option C: \( E, G, D, A, C, B, F \)**  
   - This path also visits all vertices exactly once, so it is a **valid Hamilton path**.

4. **Option D: \( B, F, C, A, D, G, E \)**  
   - This path visits all vertices exactly once, so it is a **valid Hamilton path**.

### Conclusion:
The valid Hamilton paths are:
- **B** \( C, B, F, A, D, G, E \)
- **C** \( E, G, D, A, C, B, F \)
- **D** \( B, F, C, A, D, G, E \)

So the correct answers are **B, C, and D**.

Would you like more details or have any questions?  
Here are 5 related questions to expand your understanding:

1. What are the conditions for a Hamiltonian cycle?
2. How does a Hamilton path differ from an Eulerian path?
3. Can all graphs have Hamilton paths? Why or why not?
4. What is the significance of Hamilton paths in graph theory applications?
5. How can algorithms like backtracking help in finding Hamilton paths?

**Tip:** A Hamiltonian path doesn't require you to revisit any vertex, while a Hamiltonian cycle returns to the starting vertex.

Determine two different Hamilton paths in the following graph with vertices A, B, C, D, E, F, and G. Select all options that are valid Hamilton paths.

Explore how to determine Hamilton paths in a graph with vertices A, B, C, D, E, F, and G. Learn the criteria for Hamilton paths and identify valid solutions. Suitable for advanced high school students (Grades 11-12).

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