  Let $T$ be a tournament with nondecreasing score sequence $R$ and $A$ be its
tournament matrix. An upset of $T$ corresponds to an 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 $U_{\min}(R)$ denote the set of
tournament 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
$|U_{\min}(R)|=1$. In this article, we characterize all feasible score
sequences $R$ with $|U_{\min}(R)|=1$ and give an explicit formula for the
number of the feasible score sequences $R$ with $|U_{\min}(R)|=1$.
