On the structure of tournaments consisting of only kings

The structure of some classes of tournaments consisting of only kings and their number are considered, and it is also shown that such tournaments are not simple. A tournament is called regular if all its vertices have the same entry and exit degrees. The vertex $v$ of the tournament is king if the length of the path from $v$ to any other vertex is no more than 2. The following main result has been obtained: tournament $T$ of dimension $n$, where $n$ is odd, consisting of only kings, $k$ rows of the distance matrix of which have the form $(1^{(n-1)/2},2^{(n -1)/2})$, and the rest $(1^{(n-i-1)},2^i )$, where $i\in\{1,\ldots,(n-k)/2,(n+k)/2-1,\ldots,n-2\}$, has the following structure: at the base there is a regular tournament of dimension $k$, to which two vertices $v_1$ and $v_2$ are sequentially added as follows: arcs lead from vertex $v_1$ to all vertices of tournament $T^*$; from each vertex of $T^*$ there are arcs leading to vertex $v_2$; and there is also an arc from $v_2$ to $v_1$, where $T^*$ is the tournament obtained at the previous step, until the dimension of the resulting tournament is equal to $n$. Such a tournament is not simple, and for each $n$ there are as many such tournaments as there are regular tournaments with the number of vertices equal to that of the tournament at the base.