Every strong digraph has a spanning strong subgraph with at most n+2α−2 arcs

Answering a question of Adrian Bondy (Per. Comm.), we prove that every strong digraph has a spanning strong subgraph with at most n+2α−2 arcs, where α is the size of a maximum stable set of D. Such a spanning subgraph can be found in polynomial time. An infinite family of oriented graphs for which this bound is sharp was given by Odile Favaron (Discrete Math. 146 (1995) 289). A direct corollary of our result is that there exists 2α−1 directed cycles which span D. Tibor Gallai (Theory of Graphs and its Applications, Czech. Acad. Sci. Prague, 1964, p. 161) conjectured that α directed cycles would be enough.