# Minimum transitive dominating subtournament

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 $$k=2$$.