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

Authors:
Stéphane Bessy;Stéphan Thomassé
Affiliations:
Laboratoire LaPCS, U.F.R. de Mathématiques, Université Claude Bernard, Lyon 1, 50, avenue Tony Garnier, Bâtiment RECHERCHE [B], Domaine de Gerland Lyon 69367, Cedex 07, France;Laboratoire LaPCS, U.F.R. de Mathématiques, Université Claude Bernard, Lyon 1, 50, avenue Tony Garnier, Bâtiment RECHERCHE [B], Domaine de Gerland Lyon 69367, Cedex 07, France
Venue:
Journal of Combinatorial Theory Series B
Year:
2003

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Abstract

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.