Certainly! Let's solve Problem 1 using Kruskal's algorithm to find the Minimum Spanning Tree (MST).

Kruskal's Algorithm

Step 1: List all edges and their weights in ascending order.

Here are the edges:

• $a_1 \to a_6$ with weight 1
• $a_1 \to a_2$ with weight 2
• $a_4 \to a_3$ with weight 3
• $a_6 \to a_4$ with weight 4
• $a_1 \to a_3$ with weight 5
• $a_4 \to a_5$ with weight 6
• $a_6 \to a_5$ with weight 7
• $a_3 \to a_2$ with weight 8

Step 2: Start adding edges from the list to the MST, ensuring no cycles form.

1. Add edge $a_1 \to a_6$ with weight 1.

   • Vertices connected: $\{a_1, a_6\}$.

2. Add edge $a_1 \to a_2$ with weight 2.

   • Vertices connected: $\{a_1, a_2, a_6\}$.

3. Add edge $a_4 \to a_3$ with weight 3.

   • Vertices connected: $\{a_1, a_2, a_3, a_4, a_6\}$.

4. Add edge $a_6 \to a_4$ with weight 4.

   • Vertices connected: $\{a_1, a_2, a_3, a_4, a_6\}$ (still valid since no cycle forms).

5. Add edge $a_4 \to a_5$ with weight 6.

   • Vertices connected: $\{a_1, a_2, a_3, a_4, a_5, a_6\}$.

Step 3: Check if all vertices are included in the MST.

• All vertices $\{a_1, a_2, a_3, a_4, a_5, a_6\}$ are included.

Final MST Edges:

• $a_1 \to a_6$ with weight 1
• $a_1 \to a_2$ with weight 2
• $a_4 \to a_3$ with weight 3
• $a_6 \to a_4$ with weight 4
• $a_4 \to a_5$ with weight 6

Minimum Spanning Tree Weight: The MST includes the edges with weights 1, 2, 3, 4, and 6, which adds up to 16.

Would you like more details on any of the steps? Here are five related questions to consider:

1. How do you detect cycles when using Kruskal's algorithm?
2. What is the difference between Prim's and Kruskal's algorithms in terms of their efficiency?
3. Can Kruskal's algorithm be applied to directed graphs?
4. What happens if two edges have the same weight in Kruskal's algorithm?
5. How does Kruskal's algorithm perform on dense graphs compared to sparse ones?

Tip: When applying Kruskal’s algorithm, always ensure that the edge selection doesn’t create a cycle. Using a union-find (disjoint-set) structure can make this process efficient.