Twin domination number of Tournaments

Let $D=(V,A)$ be a digraph. A subset $S$ of $V$ is called a twin dominating set of $D$ if for every vertex $v\in V-S$, there exists vertices $u_1,u_2 \in S$ such that $(v,u_1)$ and $(u_2,v)$ are arcs in $D$. The minimum cardinality of a twin dominating set in $D$ is called the twin domination number of $D$ and is denoted by $\gamma ^{*}(D)$. The upper orientable twin domination number of a graph $G$ is $DOM^{*}(G)=\max\{ \gamma ^{*}(D)|D \ \text{is an orientation of G} \}.$ It has been conjectured that for the complete graph $K_n$ with $n\geq 8$, $DOM^{*}(K_n)=\left\lceil \frac{n+1}{2}\right\rceil$. In this work we prove $DOM^{*}(K_8)= DOM^{*}(K_9)= 4$ and establish new upper bounds for $DOM^{*}(K_n)$, disproving the same above conjecture for all $n \geq 8$.