Weak covering relations

This paper proposes and justifies a natural way to weaken the concept of covering relation defined on a finite tournament. Various weak covering relations, calledk-covering relations, are introduced. To eachk-covering relation corresponds a strong uncovered set containing all nonk-covered outcomes. It is proved that those strong uncovered sets may be empty. Moreover, the set of all tournaments having an empty strong uncovered set is characterized within two rather large classes of tournaments. Finally, we offer a complete study of the cases where the directed graph defined by ak-covering relation coincides with the initial tournament.