Making a tournament k $k$‐strong

Abstract

A digraph is k ${\bf{k}}$ -strong if it has n ≥ k + 1 $n\ge k+1$ vertices and every induced subdigraph on at least n − k + 1 $n-k+1$ vertices is strongly connected. A tournament is a digraph with no pair of nonadjacent vertices. We prove that every tournament on n ≥ k + 1 $n\ge k+1$ vertices can be made k $k$ -strong by adding no more than k + 1 2 $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ arcs. This solves a conjecture from 1994. A digraph is semicomplete if there is at least one arc between any pair of distinct vertices x , y $x,y$ . Since every semicomplete digraph contains a spanning tournament, the result above also holds for semicomplete digraphs. Our result also implies that for every k ≥ 2 $k\ge 2$ , every semicomplete digraph on at least 3 k − 1 $3k-1$ vertices can be made k $k$ -strong by reversing no more than k + 1 2 $\left(\genfrac{}{}{0ex}{}{k+1}{2}\right)$ arcs.