A Special Type Of Score Sequences

Let T be a tournament with nondecreasing score sequence R(i.e.for every ri,rj in R,if i<j,then ri?rj)and A be its tournament matrix.An upset of T corresponds to an non-zero entry above the main diagonal of A.Given a feasible score sequence R,Fulkerson(1965)gave a simple recursive construction for a tournament with score sequence R and the minimum number of upsets,and Hacioglu et al.(2019)provided a construction for all of such tournament matrices.Let Umin(R)denote the set of tour-nament matrices with score sequence R that have minimum number of upsets.Brauldi and Li(1983)characterized the strong score sequences R(R is strong if a tournament T with score sequence R is strongly connected)with |Umin(R)|=1.In this article,we characterize all feasible score sequences R with |Umin(R)|=1 and give an explicit formula for the number of the feasible score sequences R with |Umin(R)|=1.