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The problem asks to apply the repeated nearest neighbor algorithm to the given weighted graph, starting and ending at vertex \( A \).

The repeated nearest neighbor algorithm works as follows:
1. Start at the initial vertex (in this case \( A \)).
2. From the current vertex, choose the edge with the smallest weight to an unvisited vertex.
3. Repeat step 2 until all vertices are visited.
4. Return to the starting vertex to complete the tour.

### Step-by-step solution:

1. **Start at vertex \( A \):**
   - The edges from \( A \) are:
     - \( A \to B \) with weight 14
     - \( A \to E \) with weight 10
     - \( A \to D \) with weight 12
   - The smallest weight is 10, so the next vertex is \( E \).

2. **Move to vertex \( E \):**
   - The edges from \( E \) are:
     - \( E \to A \) with weight 10 (already visited)
     - \( E \to D \) with weight 3
     - \( E \to C \) with weight 15
     - \( E \to B \) with weight 10
   - The smallest weight is 3, so the next vertex is \( D \).

3. **Move to vertex \( D \):**
   - The edges from \( D \) are:
     - \( D \to A \) with weight 12 (already visited)
     - \( D \to E \) with weight 3 (already visited)
     - \( D \to C \) with weight 8
     - \( D \to B \) with weight 15
   - The smallest weight is 8, so the next vertex is \( C \).

4. **Move to vertex \( C \):**
   - The edges from \( C \) are:
     - \( C \to A \) with weight 5
     - \( C \to B \) with weight 11
     - \( C \to D \) with weight 8 (already visited)
     - \( C \to E \) with weight 15 (already visited)
   - The smallest weight is 5, so the next vertex is \( A \).

Since we are back at vertex \( A \) and have visited all vertices, the tour is complete.

### Final Tour:
The path is \( A \to E \to D \to C \to A \), with a total distance of \( 10 + 3 + 8 + 5 = 26 \).

Do you have any questions, or would you like further details on any step?

### Related Questions:
1. How does the nearest neighbor algorithm compare to other algorithms like the Christofides or Held-Karp?
2. What is the total cost of the tour using this algorithm?
3. Is the nearest neighbor algorithm guaranteed to give the optimal solution?
4. How does the choice of the starting vertex affect the solution?
5. How can we adapt this algorithm to work for larger graphs?

**Tip:** The nearest neighbor algorithm is a greedy algorithm and doesn't always guarantee the optimal solution but is often used due to its simplicity and speed.

Apply the repeated nearest neighbor algorithm to the graph above. Give your answer as a list of vertices, starting and ending at vertex A.

Learn how to solve a graph theory problem using the nearest neighbor algorithm. The problem involves a weighted graph with vertices A, B, C, D, and E. A step-by-step approach is provided, demonstrating how to select the smallest edge weight at each step, returning to the starting point at A.

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