A note on decompositions of transitive tournaments

For any positive integer n, we determine all connected digraphs G of size at most four, such that a transitive tournament of order n is G-decomposable. Among others, these results disprove a generalization of a theorem of Sali and Simonyi [Orientations of self-complementary graphs and the relation of Sperner and Shannon capacities, European J. Combin. 20 (1999), 93–99].