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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?
If we add an edge to a biconnected graph with k strongly connected components, then there are three scenarios: the endpoints of the edge lie in different strongly connected component and there is no path between the two 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
Edges and graphs are emphasized.