A digraph is k ${\bf{k}}$ -strong if it has n ≥ + 1 $n\ge k+1$ vertices and every induced subdigraph on at least − $n-k+1$ strongly connected. tournament a with no pair of nonadjacent vertices. We prove that can be made $k$ by adding more than 2 $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ arcs. This solves conjecture from 1994. semicomplete there one arc between any distinct x , y $x,y$ . Since ...