The solution below got cut off. Please let me know what is the total solution:

problem:
The number of strongly connected components in a graph G is k. By how much can this number change if we add a new edge?

solution:

Suppose if we add an edge to a biconnected graph with k strongly connected components, then there are 3-situations: the endpoints of edge lie in different strongly connected component and there is no path between 2 in the original graph, the endpoints of the edge lie in different strongly connected component and there is a path between the two in the original graph, and the endpoints of the edge lie in the same strongly connected component. In the former, the edge becomes a bridge, and thus the strongly connected components remain separate. In the middle case, the edge completes a simple cycle that includes the path between the two stongly connected components and thus the number of strongly connected components decreaseby n − 1, where n is the number of strongly connected components on that path (including the two in which the endpoints