  For all positive even integers $n$, graphs of order $n$ with degree sequence
\begin{equation*} S_{n}:1,2,\dots,n/2,n/2,n/2+1,n/2+2,\dots,n-1 \end{equation*}
naturally arose in the study of a labeling problem in \cite{IMO}. This fact
motivated the authors of the aforementioned paper to study these sequences and
as a result of this study they proved that there is a unique graph of order $n$
realizing $S_{n}$ for every even integer $n$. The main goal of this paper is to
generalize this result.
