Indecomposable tournaments with minimum Slater index

The Slater index (resp. decomposability index) of a tournament is the minimum number of arcs that must be reversed in that tournament in order to make it a total order (resp. indecomposable (under modular decomposition)). The first author [H. Belkhechine, Decomposability index of tournaments, Discrete Math. 340 (2017) 2986–2994] showed that for every integer n≥5, the decomposability index of the n-vertex total order equals ⌈n+1 4 ⌉. It follows that the Slater index of an indecomposable n-vertex tournament is at least ⌈n+1 4 ⌉. This led A. Boussaïri to ask the following question during the thesis defense of the second author on July 2, 2021: what are the indecomposable tournaments T whose Slater index is minimum over all indecomposable tournaments with the same vertex set as T ? These tournaments are then the indecomposable tournaments T obtained from a total order by reversing exactly ⌈v(T )+1 4 ⌉arcs, where v(T) is the number of vertices of T . In this paper, we characterize such tournaments by means of so-called irreducible pairings. 2020 Mathematics Subject Classification. 05A18, 05C09, 05C20, 05C35, 05C75, 06A05.