Strongly Connected Components

Nodes are placed on [[stack]] in the order which they are visited
when the DFS search recursively visits a node v and its desedants, those nodes are not neccessarily popped from the stack when recursive call returns.
the node remains on the stack after it has been visited if and only if there exists a path in the input graph from it to some node earlier on the stack.
In other words, it means that in the #dfs a node would be only removed from the stack after all its connected paths has been traversed.
when the #dfs will backtrack it would remove the node on a single path and return to the root in order to start a new path.
At the end of the call that visits v and its descendants, we know whether v itself has a path to any node ealier on the stack. If so, the call returns, leaving v on the stack to resever the invariant. If not, then v must be the root of its Strongly connected components, which consists of v together with any nodes later on stack than v.
the connected component root at v is then popped from the stack and returned, again preserving the invariant.

Each node v is assigned a unique integer v.index, which numbers the nodes consecutively in the order in which they are discovered.
It also maintains a value v.lowlink that presents the smallest indx of any node on the stack know to be reachable from v through v's #dfs subtree, including v itself.
therefore v must be left the stack if v.lowlink < v.index, whereares v must be removed as the root of Strongly connected components if v.lowlink == v.index.
The value v.lowlink is computed during the #dfs from v, as this finds the node are reachable from v.