To solve the problem using Kruskal's algorithm for the given graph, follow these steps:

Steps of Kruskal's Algorithm:

1. Sort the edges in ascending order of their weights.
2. Start adding edges to the Minimum Spanning Tree (MST), ensuring no cycles are formed.
3. Stop when the MST contains $V-1$ edges (where $V$ is the number of vertices).

Extracted Data from the Graph:

The vertices are $A, B, C, D, E$, and the edges with weights are as follows:

• $AB: 26, AC: 42, AD: 16, AE: 18$
• $BC: 10, BD: 43, BE: 2$
• $CD: 8, CE: 35$
• $DE: 45$

Sorted Edges by Weight:

1. $BE: 2$
2. $CD: 8$
3. $BC: 10$
4. $AD: 16$
5. $AE: 18$
6. $AB: 26$
7. $CE: 35$
8. $AC: 42$
9. $BD: 43$
10. $DE: 45$

Building the MST:

We proceed edge by edge:

1. Add $BE: 2$ (No cycle formed).
2. Add $CD: 8$ (No cycle formed).
3. Add $BC: 10$ (No cycle formed).
4. Add $AD: 16$ (No cycle formed).

At this point, the MST has 4 edges, which is $V-1 = 5-1 = 4$, so we stop.

Total Cost of the MST:

The total cost is: $2 + 8 + 10 + 16 = 36$

Final Answer:

The total cost of the Minimum Spanning Tree is 36.

Would you like a breakdown of each step or further explanation of Kruskal's algorithm?

5 Related Questions:

1. How does Kruskal’s algorithm differ from Prim’s algorithm for finding an MST?
2. What happens if there are multiple edges with the same weight in Kruskal's algorithm?
3. How can the union-find data structure optimize Kruskal’s algorithm?
4. What is the time complexity of Kruskal’s algorithm?
5. Can Kruskal’s algorithm work on disconnected graphs?

Tip:

When using Kruskal’s algorithm, always double-check edge selection to avoid forming cycles, which can invalidate the MST.