Consider the cities E, F, G, H, I, J. The costs of the possible roads between cities are given below: c(E,G) = 13 c(E, H) = 12 c(E,I) = 14 c(F,G) = 15 c(G, H) = 8 c(H, I) = 9 c(H, J) = 10 c(I, J) = 11 What is the minimum cost to build a road system that connects all the cities?

The given problem involves determining the minimum cost to connect all the cities \( E, F, G, H, I, J \) using the provided road costs. This problem can be solved using **Kruskal's Algorithm**, a greedy algorithm for finding the **Minimum Spanning Tree (MST)** of a graph, which ensures that all cities are connected with the least total road cost.

### Step-by-Step Approach:

1. **List the cities**: The cities are \( E, F, G, H, I, J \).
2. **List the road costs**: 
   - \( c(E, G) = 13 \)
   - \( c(E, H) = 12 \)
   - \( c(E, I) = 14 \)
   - \( c(F, G) = 15 \)
   - \( c(G, H) = 8 \)
   - \( c(H, I) = 9 \)
   - \( c(H, J) = 10 \)
   - \( c(I, J) = 11 \)

3. **Sort the roads by cost**: 
   - \( c(G, H) = 8 \)
   - \( c(H, I) = 9 \)
   - \( c(H, J) = 10 \)
   - \( c(I, J) = 11 \)
   - \( c(E, H) = 12 \)
   - \( c(E, G) = 13 \)
   - \( c(E, I) = 14 \)
   - \( c(F, G) = 15 \)

4. **Apply Kruskal’s Algorithm**:
   - Start with no roads and no connected cities.
   - Add roads one by one in increasing order of cost, ensuring that no cycle is formed.

   - Add \( c(G, H) = 8 \) → Cities connected: \( G \)-\( H \)
   - Add \( c(H, I) = 9 \) → Cities connected: \( G \)-\( H \)-\( I \)
   - Add \( c(H, J) = 10 \) → Cities connected: \( G \)-\( H \)-\( I \)-\( J \)
   - Add \( c(I, J) = 11 \) → This forms a cycle, so **skip it**.
   - Add \( c(E, H) = 12 \) → Cities connected: \( E \)-\( G \)-\( H \)-\( I \)-\( J \)
   - Add \( c(E, G) = 13 \) → This forms a cycle, so **skip it**.
   - Add \( c(E, I) = 14 \) → This forms a cycle, so **skip it**.
   - Add \( c(F, G) = 15 \) → Cities connected: \( F \)-\( E \)-\( G \)-\( H \)-\( I \)-\( J \)

Now all cities \( E, F, G, H, I, J \) are connected.

### Total Minimum Cost:
The total cost of the roads included in the minimum spanning tree is:

\[
8 + 9 + 10 + 12 + 15 = 54
\]

Thus, the minimum cost to build a road system that connects all the cities is **54**.

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Would you like more details on how Kruskal's algorithm works or have any questions?

### Related Questions:
1. What is Prim's algorithm, and how does it differ from Kruskal's algorithm?
2. How do you detect a cycle in a graph when building a spanning tree?
3. How can the Union-Find data structure optimize Kruskal's algorithm?
4. What is the significance of spanning trees in network design?
5. How can Kruskal's algorithm be applied in real-world transportation or logistics planning?

**Tip**: Kruskal’s algorithm is efficient for sparse graphs, where the number of edges is relatively small compared to the number of nodes.

Find the minimum cost to connect cities E, F, G, H, I, and J using Kruskal's Algorithm. Learn step-by-step how to sort road costs and apply cycle detection to ensure the least expensive road system. Explore the concept of Minimum Spanning Trees, Kruskal’s Algorithm, and its application in network design.

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