Suppose we take all MSTs of a graph and build a new graph where each vertex corresponds to a MST of the first graph and two vertices are connected if their corresponding spannings trees differ by only one edge.

Is this second graph make of a single component?

I am guessing it is, but guessing is not prooving...

Because any cut of the graph will give MSTs that differ by one edge - one of the edges that cross the cut with minimum weight.

Is there a name for this problem?