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 C1 k / log k ≤ F ( k ) ≤ C2 k log k for suitable positive constants C1 and C2.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Dominating set