The existence and uniqueness of strong kings in tournaments

A king x in a tournament T is a player who beats any other player y directly (i.e., x->y) or indirectly through a third player z (i.e., x->z and z->y). For x,[email protected]?V(T), let b(x,y) denote the number of third players through which x beats y indirectly. Then, a king x is strong if the following condition is fulfilled: b(x,y)>b(y,x) whenever y->x. In this paper, a result shows that for a tournament on n players there exist exactly k strong kings, 1=