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.