如何找到最小的边成本以添加到加权图中以使其双向连接？

Question

This is related to the following problem- http://www.iarcs.org.in/inoi/contests/aug2005/Advanced-2.php . Any hints on how to solve this problem?

Answer 1

Here are a few simple observation that give us a solution to this problem:

1. If a non-capital city is removed, the rest of the graph is still connected. It is the case because we can reach one city from another through the capital.

2. If a capital is removed, the rest of the graph is empty.

It means that we just need to find a minimum spanning tree in the given graph after removing the first vertex and all edges adjacent to it. You can find a minimum spanning tree using any standard algorithm(for example, Prim's).