For the complete digraph with n⩾3, its half as well as its third (or near-third) part, both non-self-converse, are exhibited. A backtracking method for generating a tth part of a digraph is sketched. It is proved that some self-converse digraphs are not among the near-third parts of the complete digraph in four of the six possible cases. For a third part D of is found such that D is a self-converse oriented graph and all D-decompositions of have trivial automorphism group.