A Recurrence for Bounds on Dominating Sets in k-Majority Tournaments

A k-majority tournament is realized by 2k−1 linear orders on the set of vertices, where a vertex u dominates v if u precedes v in at least k of the orders. Various properties of such tournaments have been studied, among them the problem of finding the size of a smallest dominating set. It is known that 2-majority tournaments are dominated by 3 vertices and that k-majority tournaments are dominated by O(k log k) vertices. However, precise values are not known even for k = 3. We establish new upper bounds for the size of a smallest dominating set in k-majority tournaments that considerably improve upon previous bounds for small k. In particular our result shows that 3-majority tournaments are dominated by at most 12 vertices.