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.