https://file.aiquickdraw.com/mathbot/file/1730180910518nj3g6rlo.jpg

To solve this problem, we’ll use the **Cheapest Link Algorithm** (also known as the Best Edge Algorithm). The goal is to construct a tour that starts and ends at vertex \( A \), visiting each vertex exactly once with the minimum possible edge cost. Here’s the step-by-step approach:

### Step 1: Identify the Cheapest Edges
List the edges in ascending order based on their weights:
- **1**: \( A \) to \( E \)
- **3**: \( D \) to \( E \)
- **4**: \( B \) to \( A \)
- **6**: \( B \) to \( C \)
- **8**: \( B \) to \( A \)
- **9**: \( C \) to \( D \)
- **10**: \( C \) to \( D \)
- **13**: \( B \) to \( D \)
- **14**: \( A \) to \( D \)
- **15**: \( C \) to \( E \)

### Step 2: Construct the Tour Using Cheapest Edges
Start adding the edges in order of increasing weight, ensuring:
1. Each vertex has no more than two edges connected to it (to avoid loops before forming a complete tour).
2. We do not complete a cycle until we have included all vertices.

#### Applying the Steps:
1. Add \( A \)-\( E \) (weight 1).
2. Add \( D \)-\( E \) (weight 3).
3. Add \( B \)-\( A \) (weight 4).
4. Add \( B \)-\( C \) (weight 6).

At this point, every vertex has exactly two edges, creating a complete tour without any extra connections.

### Step 3: List the Tour
Starting and ending at \( A \), we get:
**A → E → D → C → B → A**

### Final Answer
The tour with the minimum cost is:
\[
A \rightarrow E \rightarrow D \rightarrow C \rightarrow B \rightarrow A
\]

Would you like further explanation on any of the steps? 

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**Related Questions:**
1. What would be the result if we started at a different vertex, like \( B \), instead of \( A \)?
2. Can this algorithm be used if the graph had directed edges?
3. How would the solution change if we had more vertices?
4. Is the Cheapest Link Algorithm optimal for all graphs?
5. What are the advantages and disadvantages of the Cheapest Link Algorithm compared to other algorithms like Prim's?

**Tip:** Always check that each vertex has exactly two edges in the final solution to ensure a valid Hamiltonian cycle.

Apply the best edge algorithm (also called the cheapest link algorithm) to the graph above. Give your answer as a list of vertices, starting and ending at vertex A. Example: ABCDEA.

Solve a traveling salesman problem using the Cheapest Link Algorithm, starting and ending at vertex A. This problem involves graph theory concepts and requires identifying the shortest path while visiting each vertex exactly once.

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