To solve this using the sorted edges algorithm, we aim to construct a Hamiltonian circuit (a tour visiting every vertex exactly once and returning to the start) by selecting edges in order of increasing weight, ensuring no vertex is visited more than once except the starting/ending vertex and avoiding closed loops until the full circuit is formed.

Steps:

1. Sort the edges by weight in ascending order. We list all the edges and their weights:

   • $A \to E$: 1
   • $B \to C$: 2
   • $A \to B$: 6
   • $D \to E$: 5
   • $D \to C$: 7
   • $B \to E$: 12
   • $B \to A$: 14
   • $C \to D$: 13

   Sorted: $A \to E$ (1), $B \to C$ (2), $A \to B$ (6), $D \to E$ (5), $D \to C$ (7), $B \to E$ (12), $C \to D$ (13), $B \to A$ (14).

2. Add edges without forming loops or revisiting a vertex unless completing the circuit. Following the sorted order:

   • $A \to E$ (1): Add this edge.
   • $B \to C$ (2): Add this edge.
   • $D \to E$ (5): Add this edge.
   • $A \to B$ (6): Add this edge.
   • $D \to C$ (7): Add this edge.

   At this point, we have: $A \to E \to D \to C \to B \to A$.

3. Finalize the circuit: Ensure the vertices $A, B, C, D, E$ are connected in a full circuit: Final path: $A \to E \to D \to C \to B \to A$.

Final Answer:

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Would you like a detailed explanation of any step?

Questions to expand your understanding:

1. What is the sorted edges algorithm, and how is it different from other Hamiltonian circuit algorithms?
2. How does sorting edges help simplify complex graph problems?
3. What are the rules for avoiding loops in the sorted edges algorithm?
4. Can this algorithm fail for certain graphs? If so, why?
5. How does the concept of minimum spanning trees differ from Hamiltonian circuits?

Tip: Always verify your path to ensure it visits all vertices exactly once and forms a complete circuit!