Question 8451 – Engineering-Mathematics

G(V, E) is a directed graph.

→ It strongly connected.
(A) G₁=(V,E₁) where E₁={(u,v)|(u,v)∉E}
If (u, v) does not belong to the edge set ‘E’, then it indicates there are no edges. So, it is not connected.
(B) G₂=(V,E₂ )where E₂={(u,v)│(u,v)∈E}
Given that ‘G’ is directed graph, i.e., it has path from each vertex to every other vertex.
Though direction is changed from (u, v) to (v, u), it is still connected component same as ‘G’.
(C) G₃=(V,E₃) where E₃={(u,v)|there is a path of length≤2 from u to v in E}
This can also be true.
eg:

Both from each vertex to other vertex is also exists. So it is also strongly connected graph.
(D) G₄=(V₄,E) where V₄ is the set of vertices in G which are not isolated.

If ‘G’ has same ‘x’ no. of isolated vertices, one strongly connected component
then no. of SCC = x + 1
G₄ contain only ‘1’ component, which is not same as G.