On the sizes of the graphs G, G^r, G^r\G: The directed case

Let G be a directed graph and G^r be its r-th power. We study different issues dealing with the number of arcs, or size, of G and G ^r: given the order and diameter of a strongly connected digraph, what is its maximum size, and which are the graphs achieving this bound? What is the minimum size of the r-th power of a strongly connected digraph, and which are the graphs achieving this bound? Given all strongly connected digraphs G of order n such that Gr = K_n, what is the minimum number of arcs that are added when going from G to G^r, and which are the graphs achieving this bound?.