Strongly connected components:

1
Strongly connected components
a
b
c
d
e
j
f
g
h
i
a b f
e i j
c d g h
2
The transpose of a graph, GT is one in which all
of the edges are in the reverse direction.
SCC of G SCC of GT
SCC algorithm 1. Call DFS(G) to compute
finishing times fv for each vertex 2. Compute
GT 3. Call DFS(GT), but in the main loop of DFS,
consider the vertices in order of decreasing
fv, as computed in DFS(G) 4. Output the
vertices of each tree in the depth-first forest
from DFS(GT) as a separate SCC
O (n m) time
3
Lemma 23.12 If two vertices are in the same
strongly connected component, then no path
between them ever leaves the SCC.
Theorem 23.13 In any DFS, all vertices in the
same SCC are placed in the same depth-first tree.
The forefather f(u) of a vertex u is the vertex w
that is reachable from u that finished last in
the DFS of step 1.