To solve the problem of finding the Minimum Spanning Tree (MST) using Kruskal's Algorithm, we follow these steps systematically. Below is a detailed explanation based on the given weighted graph:

Step-by-Step Process

1. Sort all edges in non-decreasing order of their weights: First, list all the edges and their weights. Then sort them in ascending order of weight:

   Edge (Weight):
   E-F (2), D-E (5), B-E (7), A-B (4), A-D (10), C-D (11),
   C-F (9), F-G (14), C-G (15)
   

   After sorting by weight:

   E-F (2), D-E (5), A-B (4), B-E (7), C-F (9),
   A-D (10), C-D (11), F-G (14), C-G (15)
   

2. Initialize a forest (each node is its own tree): Each vertex is initially treated as a separate component: {A}, {B}, {C}, {D}, {E}, {F}, {G}.

3. Add edges one by one, following these rules:

   • Add the edge with the smallest weight that does not form a cycle.
   • Use a union-find data structure to check for cycles (union operation for connecting components and find operation for cycle detection).

4. Build the MST step-by-step:

   • Add E-F (2): This is the smallest weight. No cycle is formed. The MST now includes: {E, F}. Total weight = 2.

   • Add D-E (5): No cycle is formed. The MST now includes: {D, E, F}. Total weight = 2 + 5 = 7.

   • Add A-B (4): No cycle is formed. The MST now includes: {A, B}, {D, E, F}. Total weight = 7 + 4 = 11.

   • Add B-E (7): No cycle is formed. The MST now includes: {A, B, E, F, D}. Total weight = 11 + 7 = 18.

   • Add C-F (9): No cycle is formed. The MST now includes: {A, B, C, D, E, F}. Total weight = 18 + 9 = 27.

   • Add A-D (10): Adding this edge would form a cycle, so skip it.

   • Add C-D (11): Adding this edge would form a cycle, so skip it.

   • Add F-G (14): No cycle is formed. The MST now includes all vertices: {A, B, C, D, E, F, G}. Total weight = 27 + 14 = 41.

   • Stop here: All vertices are included in the MST.

5. Final MST: The Minimum Spanning Tree includes the edges:

   E-F (2), D-E (5), A-B (4), B-E (7), C-F (9), F-G (14)
   

   Total weight of the MST = 41.

Summary of the Steps in Kruskal's Algorithm

1. Sort all edges by weight.
2. Start with an empty MST and treat each node as a separate tree.
3. Add the smallest edge that does not form a cycle (use union-find).
4. Repeat until all nodes are connected or there are $n-1$ edges in the MST (where $n$ is the number of vertices).

Would you like further explanation of any step or details about cycle detection? Let me know!

Follow-up Questions

1. Why does Kruskal's Algorithm always produce a Minimum Spanning Tree for a connected graph?
2. How does a union-find data structure help in detecting cycles in Kruskal's Algorithm?
3. What would happen if we used a greedy approach but ignored the cycle-checking condition?
4. Can Kruskal’s Algorithm be used for directed graphs? Why or why not?
5. How does the complexity of Kruskal’s Algorithm compare to Prim’s Algorithm?

Tip: When solving MST problems, always verify that your graph is connected. If it isn’t, there will be no spanning tree!