For any positive integer $ k$ , does there exist a tournament such that the smallest dominating set, which also forms a transitive subtournament, has size exactly $ k$ ?

A tournament that does *not* work for $ k>2$ is one where the vertices are on a cycle and each vertex has an edge to $ (n-1)/2$ following vertices clockwise — in this case taking two opposite vertices already gives a dominating set which is also transitive, meaning $ k=2$ .