To determine the minimal spanning tree (MST) for this graph, we’ll analyze the edges based on their weights and select the smallest edges without forming any cycles. The goal is to include each vertex with the minimum possible weight.

Step-by-Step Solution

1. Identify the Edge with Minimum Weight:

   • We start with the smallest weights. From the image, the edges with the smallest weights are as follows:
     • Edge $(1, 2)$ with weight 2
     • Edge $(4, 6)$ with weight 3
     • Edge $(3, 5)$ with weight 3

2. Continue Adding the Next Smallest Edges While Avoiding Cycles:

   • After adding the edges $(1, 2)$, $(4, 6)$, and $(3, 5)$, we move to the next smallest edges:
     • Edge $(1, 3)$ with weight 4
     • Edge $(2, 5)$ with weight 5
     • Edge $(3, 4)$ with weight 5

3. Construct the MST:

   • By selecting these edges and checking to avoid any cycles, we get the following edges in our MST: $A = \{(1, 2), (2, 5), (1, 3), (4, 6), (3, 5), (3, 4)\}$

4. Match with Options:

   • This set corresponds to option (b): $A = \{(1, 2), (2, 5), (1, 3), (4, 6), (3, 4), (3, 5)\}$

Final Answer

The correct answer is (b).

Would you like more details on the method used here, such as Kruskal's or Prim's algorithm?

Here are five related questions that can expand your understanding:

1. How does Kruskal's algorithm work for finding a minimal spanning tree?
2. What is Prim's algorithm, and how does it differ from Kruskal's?
3. Why do we avoid cycles when constructing a minimal spanning tree?
4. How would the solution change if we had a graph with different weights?
5. Can a minimal spanning tree be unique?

Tip: Always check the total weight of the MST to verify the minimal path, especially if there are multiple ways to connect nodes without cycles.