https://file.aiquickdraw.com/mathbot/file/17339431476933mytovml.jpg

To find the Minimum Spanning Tree (MST) for the graph shown in the image, we will follow Kruskal's or Prim's algorithm. Here's a step-by-step explanation based on Kruskal’s algorithm:

### Steps for Kruskal's Algorithm:

1. **List all edges and their weights**:  
   From the image, we can extract the edges and weights:
   ```
   a-b: 2, a-d: 2, a-e: 4, a-c: 5
   b-e: 4, b-c: 3
   c-f: 5, c-e: 3, c-g: 1
   d-e: 1, d-h: 4, d-i: 4
   e-h: 2, e-f: 3
   f-g: 2
   h-i: 2, h-g: 4
   ```

2. **Sort edges by weight** (smallest to largest):
   ```
   d-e: 1, c-g: 1
   a-b: 2, a-d: 2, f-g: 2, h-i: 2, e-h: 2
   b-c: 3, c-e: 3, e-f: 3
   a-e: 4, d-h: 4, d-i: 4, h-g: 4, b-e: 4
   a-c: 5, c-f: 5
   ```

3. **Add edges to the MST** ensuring no cycles are formed:
   - Start with the smallest edge:
     - Add **d-e (1)** to the MST.
     - Add **c-g (1)** to the MST.
   - Add edges with weight 2:
     - Add **a-b (2)** to the MST.
     - Add **a-d (2)** to the MST.
     - Add **f-g (2)** to the MST.
     - Add **h-i (2)** to the MST.
     - Add **e-h (2)** to the MST.
   - At this point, all nodes are connected, and we stop adding edges.  

4. **MST edges**:
   ```
   d-e, c-g, a-b, a-d, f-g, h-i, e-h
   ```

5. **Total weight of MST**:  
   ```
   1 + 1 + 2 + 2 + 2 + 2 + 2 = 12
   ```

### Final MST:
The Minimum Spanning Tree includes the edges:
- \(d-e\), \(c-g\), \(a-b\), \(a-d\), \(f-g\), \(h-i\), \(e-h\)  
with a total weight of **12**.

---

Would you like me to explain Prim's algorithm for this graph or provide a visual representation? Let me know!  

### Related Questions:
1. What is the difference between Kruskal's and Prim's algorithms for MST?
2. How does the choice of starting node affect Prim's algorithm?
3. Can we solve this MST problem using adjacency matrix representation?
4. What is the time complexity of Kruskal’s algorithm in this scenario?
5. How would adding or removing an edge impact the MST of this graph?

**Tip**: Always double-check for cycles when applying Kruskal’s algorithm to avoid including unnecessary edges.

Learn how to find the Minimum Spanning Tree (MST) for a weighted graph using Kruskal's algorithm. This solution explains sorting edges by weight, avoiding cycles with union-find, and achieving the optimal total weight of 12. Suitable for Grades 9-12.

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