Enumeration of indecomposable tournaments with minimum Slater index

The Slater index of a tournament is the minimum number of arcs that must be reversed in that tournament to make it a total order. Let Formula: see text be an integer with Formula: see text 5. The minimum value of the Slater index over the indecomposable Formula: see text-vertex tournaments is Formula: see text. In a recent paper H. Belkhechine, C. Ben Salha, R. Romdhane, Indecomposable tournaments with minimum Slater index, Accepted and to appear soon in Adv. Pure Appl. Math., we characterized the indecomposable n-vertex tournaments with at least five vertices and minimum Slater index, i.e., with Slater index Formula: see text. In this paper, we asymptotically enumerate these tournaments.