# Transitive Closure of a graph

Assuming we have a DAG, $$G = (V, E)$$, and we know that we can calculate $$G$$'s transitive closure in time complexity of $$f(|V|, |E|)$$, whereas $$f$$ is monotonic increasing function.
Show that given a directed graph (not necessarily DAG) $$G^{'}$$, we can calculate it's transitive closure, $$G^{+}=(V, E^{+})$$, in time complexity of $$O(f(|V|, |E|) + |V| + |E^{+}|)$$.
We can assume all the graphs are represented in adjacency matrices.
I don't have any idea for this one. I tried to read stuff about this topic, including in cormen's book, but it didn't lead me to any solution.

Using Kosaraju's algorithm, you can compute the strongly connected components (SCC) of $$G$$ in linear time.
Now to compute $$G^+$$:
• if $$u$$ and $$v$$ are in the same SCC, that means there is an edge from $$u$$ to $$v$$ and from $$v$$ to $$u$$;
• if $$u\in C$$ and $$v\in C'$$ where $$C$$ and $$C'$$ are two different SCC such that there is an edge from $$C$$ to $$C'$$ in the transitive closure of the metagraph, that means there is an edge from $$u$$ to $$v$$ (but not from $$v$$ to $$u$$).