The classic Mininum Spanning Tree (MST) algorithms can be modified to find the Maximum Spanning Tree instead.
Can an algorithm such as Kruskal's be modified to return a spanning tree that is strictly more costly than an MST, but is the second cheapest? For example, if you switch one of the edges in this spanning tree, you end up with an MST and vice versa.
My question, though, is simply: How can I find the second cheapest spanning tree, given a graph $G$ with an MST?