Majority Digraphs

Abstract

A majority digraph is a finite simple digraph G = ( V , → ) G=(V,\to ) such that there exist finite sets A v A_v for the vertices v ∈ V v\in V with the following property: u → v u\to v if and only if “more than half of the A u A_u are A v A_v ”. That is, u → v u\to v if and only if | A u ∩ A v | > 1 2 ⋅ | A u | |A_u \cap A_v | > \frac {1}{2} \cdot |A_u| . We characterize the majority digraphs as the digraphs with the property that every directed cycle has a reversal. If we change 1 2 \frac {1}{2} to any real number α ∈ ( 0 , 1 ) \alpha \in (0,1) , we obtain the same class of digraphs. We apply the characterization result to obtain a result on the logic of assertions “most X X are Y Y ” and the standard connectives of propositional logic.