Making a tournament k-arc-strong by reversing arcs

We prove that every tournament T=(V, A) on n ≥ 2k + 1 vertices can be made k-arc-strong by reversing no more than k(k + l)/2 arcs. This is best possible as the transitive tournament needs this many arcs to be reversed. We show that the number of arcs we need to reverse in order to make a tournament k-arc-strong is closely related to the number of arcs we need to reverse just to achieve in- and out-degree at least k. We also discuss the relations of our results to related problems and conjectures. The digraphs in this note may have multiple arcs but no loops. In general the notation follows [1].