Transitive Closure vs Reachability in Graphs

I am facing the most curious situation with [my current information of] transitive closure algorithms. Specifically, is what follows not an algorithm for finding the transitive closure of a graph G(V, E) and is the time complexity not O(|V|+|E|):

Use Kosaraju's to compute SCCs (strongly connected components) -- O(|V| + |E|)

Create a DAG from the SCCs as (super)nodes -- O(|V| + |E|)

In a single supernode, reachability of all nodes is the same and equal to the union of the nodes in the containing supernode plus the reachable supernodes in the DAG -- O(|V| + |E|) to compute reachability in the DAG and O(|V| * |V|) total to compute reachability sets for all nodes

These reachability sets do constitute the transitive closure of the graph wholly, right?

But there aren't any O(|V| + |E|) transitive closure algorithms around.

What am I missing?

Your algorithm is known as Purdom's algorithm. There are two versions of transitive closure:

• Calculate the transitive closure as a relation. This cannot run in time $O(|V|+|E|)$, since the output could be bigger than that (consider a directed path, for example).

• Calculate enough information to be able to answer oracle queries to the transitive closure in $O(1)$. Even in this case, the algorithm doesn't seem to run in time $O(|V|+|E|)$. Hopefully the link would make it clear why.

StackExchange Computer Science Q#110250, answer score: 2