  Given a graph and an integer $k$, it is an NP-complete problem to decide
whether there is a dominating set of size at most $k$. In this paper we study
this problem for the Knodel Graph on $n$ vertices using elementary number
theory techniques. In particular, we show an explicit upper bound for the
domination number of the Knodel Graph on $n$ vertices any time that we can find
a prime number $p$ dividing $n$ for which $2$ is a primitive root.
