Question: Apply the sorted edges algorithm to the graph above. Give your answer as a list of vertices, starting and ending at vertex $A$. Example: $ABCDEA$ Graph: top-left $C$ connected to $B$ with weight $3$, to $A$ with weight $2$, to $D$ with weight $15$, and to $E$ with weight $10$; top-right $B$ connected to $A$ with weight $1$, to $E$ with weight $11$, and to $D$ with weight $12$; center-right $A$ connected to $E$ with weight $7$ and to $D$ with weight $5$; bottom-left $D$ connected to $E$ with weight $8$.

1. **State the problem:** We need to apply the sorted edges algorithm (also known as the greedy algorithm for the Traveling Salesman Problem) to find a Hamiltonian circuit starting and ending at vertex $A$. 2. **List all edges with weights and sort them in ascending order:** - $BA = 1$ - $CA = 2$ - $CB = 3$ - $AD = 5$ - $AE = 7$ - $DE = 8$ - $CE = 10$ - $BE = 11$ - $BD = 12$ - $CD = 15$ 3. **Apply the sorted edges algorithm step-by-step:** - Add edge $BA = 1$ (connects $A$ and $B$). - Add edge $CA = 2$ (connects $A$ and $C$). - Skip edge $CB = 3$ because adding it creates a cycle $A-B-C-A$ too early. - Add edge $AD = 5$ (connects $A$ and $D$). - Skip edge $AE = 7$ because vertex $A$ would have degree 3 (not allowed). - Add edge $DE = 8$ (connects $D$ and $E$). - Skip edges $CE = 10$, $BE = 11$, $BD = 12$, and $CD = 15$ because they create cycles or vertices with degree $>2$. 4. **Check degrees of vertices:** - $A$: degree 3 (edges $A-B$, $A-C$, $A-D$) — invalid for Hamiltonian circuit. - $B$: degree 1 - $C$: degree 1 - $D$: degree 2 - $E$: degree 1 5. **Adjust edges to fix degrees:** - Remove edge $A-C$ (weight 2) to reduce $A$'s degree. - Add edge $C-B$ (weight 3) to connect $C$ and $B$. Edges now: $A-B$, $B-C$, $A-D$, $D-E$. 6. **Check degrees again:** - $A$: degree 2 ($A-B$, $A-D$) - $B$: degree 2 ($A-B$, $B-C$) - $C$: degree 1 ($B-C$) - $D$: degree 2 ($A-D$, $D-E$) - $E$: degree 1 ($D-E$) 7. **Add edge $C-E$ (weight 10) to connect $C$ and $E$ and complete the circuit:** Edges: $A-B$, $B-C$, $C-E$, $E-D$, $D-A$. 8. **Form the Hamiltonian circuit starting and ending at $A$:** $$ABCDEA$$ 9. **Final answer:** The Hamiltonian circuit using the sorted edges algorithm is $ABCDEA$, visiting every vertex exactly once and returning to $A$.