Dominating sets in k-majority tournaments

Abstract

A k- majority tournament T on a finite vertex set V is defined by a set of 2 k - 1 linear orderings of V, with u → v if and only if u lies above v in at least k of the orders. Motivated in part by the phenomenon of “non-transitive dice”, we let F ( k ) be the maximum over all k-majority tournaments T of the size of a minimum dominating set of T. We show that F ( k ) exists for all k > 0 , that F ( 2 ) = 3 and that in general C 1 k / log k ≤ F ( k ) ≤ C 2 k log k for suitable positive constants C 1 and C 2 .