Strongly Connected Components

Notes

• A strongly connected component of a graph G is the maximal set of vertices, C ⊆ G, such that for every pair of vertexes u,v in C, there is a path from u to v and a path from v to u. Or u is reachable from v and v is reachable from u ($u \rightsquigarrow v , v \rightsquigarrow i$).
• In our example graph {s,v,w} and {x, z} are strongly connected components.
• For a graph, E^T is found by reversing the direction of all the edges
  • ie if (u,v) ∈ E, (v,u) ∈ E^T
• The strongly connected components of E and E^T are the same.
• This fact is used in the following algorithm

• STRONGLY-CONNECTED-COMPONENTS(G)
  
   Call DFS(G) to compute u.f for each vertex u
   Compute GT
   Call DFS(GT)  but select edges by decreasing values of u.f computed in the previous DFS algorithm.
   Output the vertices of each tree in the depth first forest formed in the last DFS search as strongly connected components.