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What if I do not use G transpose in calculating Strongly Connected Components?

The vertices in strongly connected component are, by definiton, connected to each other (by a path, not necessarily by direct edge). if you make first DFS call on vertex X, you find out "which vertices is X connected to" (X -> N). To make sure that all those vertices are connected to X (N -> X) and therefore validate strong connectivity you need to traverse the edges in reversed directions. The easiest way to do such is by transposing the graph.

If you look for proof of the algorithm, i am sure you will find some. It may not be the easiest to understand but check this source for example: Correctness of the algorithm for finding strongly connected components

Categories : Algorithm

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