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The transitive closure of a graph is defined e. g. here: http://mathworld.wolfram.com/TransitiveClosure.html
It is easily possible in O(n^3), where n is the number of vertices. I was wondering if it can be done in time O(n^2).
Nope. I don’t think there is a O(n2) algorithm for it. I would expect if such an algorithm existed, you could solve all pair shortest paths problem in O(n2) too, which is not the case. The asymptotically fastest algorithm I can think of is an implementation of Dijkstra’s shortest path algorithm with a Fibonacci heap (O(n2log n) in not very dense graphs).