To solve this problem, we need to apply the Nearest Neighbor Algorithm starting at vertex $A$. The algorithm involves selecting the edge with the smallest weight connecting to the next unvisited vertex. Here’s the step-by-step process:

Step 1: Starting at $A$

• From $A$, the possible edges are:

  • $A \to B$ (weight $5$)
  • $A \to C$ (weight $9$)
  • $A \to D$ (weight $13$)
  • $A \to E$ (weight $7$)
  • $A \to F$ (weight $2$)

• The smallest weight is $2$, so go to $F$.

• Path so far: $A \to F$.

Step 2: At $F$

• From $F$, the possible edges (excluding $A$, which is already visited) are:

  • $F \to B$ (weight $6$)
  • $F \to C$ (weight $11$)
  • $F \to D$ (weight $3$)
  • $F \to E$ (weight $1$)

• The smallest weight is $1$, so go to $E$.

• Path so far: $A \to F \to E$.

Step 3: At $E$

• From $E$, the possible edges (excluding $F$ and $A$) are:

  • $E \to B$ (weight $7$)
  • $E \to C$ (weight $8$)
  • $E \to D$ (weight $14$)

• The smallest weight is $7$, so go to $B$.

• Path so far: $A \to F \to E \to B$.

Step 4: At $B$

• From $B$, the possible edges (excluding $E$, $F$, and $A$) are:

  • $B \to C$ (weight $12$)
  • $B \to D$ (weight $6$)

• The smallest weight is $6$, so go to $D$.

• Path so far: $A \to F \to E \to B \to D$.

Step 5: At $D$

• From $D$, the only remaining unvisited vertex is $C$ (weight $10$).
• Go to $C$.
• Path so far: $A \to F \to E \to B \to D \to C$.

Step 6: Return to $A$

• From $C$, return to $A$ (weight $9$).
• Final path: $A \to F \to E \to B \to D \to C \to A$.

Final Answer:

The path is: $\boxed{AFEBDCA}$

Let me know if you'd like further clarification or help with related concepts!

Related Questions:

1. What is the total weight of the path $AFEBDCA$?
2. How does the Nearest Neighbor Algorithm compare to other algorithms like the Brute Force method?
3. Can the Nearest Neighbor Algorithm guarantee the optimal solution for all graphs?
4. What happens if you start at a different vertex, such as $B$ or $D$?
5. How would you solve this problem using a Hamiltonian circuit approach?

Tip:

When using the Nearest Neighbor Algorithm, always keep track of visited vertices to avoid loops and ensure every vertex is included.