Dominating, weakly stable, and uncovered sets: Properties and generalizations

We consider the problem of collective choice in a tournament, i.e., when the majority relation, which plays the role of the collective preference system on this set of alternatives, can be represented by a complete asymmetric oriented graph. We compare three solutions of the collective choice problem: minimal dominating, uncovered, and minimal weakly stable sets. We construct generalizations of the minimal dominating set and find out, with their help, how the system of dominating sets looks like in the general case. We formulate a criterion that determines whether an alternative belongs to a minimal weakly stable set. We find out how minimal weakly stable sets relate to uncovered sets. Based on the notion of stability of an alternative and the set of alternatives we construct generalizations for the notions of uncovered and weakly stable sets – the classes of $k$-stable alternatives and $k$-stable sets. We prove inclusion relations between these classes.