https://file.aiquickdraw.com/mathbot/file/1727887091042mhxv76p1.jpg

To solve this problem using the **sorted edges algorithm** for finding a Hamiltonian Circuit, we need to follow these steps:

### Step-by-Step Process:

1. **List all edges in ascending order of weights**:
   - (A, C) = 2
   - (C, D) = 5
   - (D, E) = 6
   - (A, E) = 4
   - (B, C) = 9
   - (A, B) = 10
   - (B, D) = 12
   - (A, D) = 14
   - (B, E) = 13
   - (C, E) = 15

2. **Start adding edges from the list, without forming a smaller circuit or exceeding the degree of 2 for any vertex.**
   - Add edge (A, C) = 2.
   - Add edge (C, D) = 5.
   - Add edge (A, E) = 4.
   - Add edge (D, E) = 6.
   - Add edge (B, C) = 9.
   
   At this point, all vertices are connected and a Hamiltonian Circuit is formed. The remaining edges would either exceed the degree of 2 for some vertices or create smaller loops.

3. **List the circuit**:
   A possible Hamiltonian circuit starting and ending at vertex A could be:
   - **A → C → B → D → E → A**.

Would you like a deeper explanation of any of the steps or any further clarifications?

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Here are 5 related questions to explore the topic further:
1. What is a Hamiltonian path, and how does it differ from a Hamiltonian circuit?
2. How does the sorted edges algorithm compare to other algorithms for finding Hamiltonian circuits?
3. Can all graphs have a Hamiltonian circuit?
4. What are some real-world applications of Hamiltonian circuits?
5. How does the degree of a vertex affect the possibility of forming a Hamiltonian circuit?

**Tip:** When using the sorted edges algorithm, always ensure you don't close a smaller cycle prematurely or exceed the degree of 2 for any vertex before all vertices are included.

Apply the sorted edges algorithm to the graph. Give your answer as a list of vertices, starting and ending at vertex A.

Learn how to apply the sorted edges algorithm to find a Hamiltonian circuit in a graph. This problem demonstrates the step-by-step process of finding a Hamiltonian circuit with edge weights, suitable for high school students in Grades 10-12.

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