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→yx→y) or indirectly through a third player z   (i.e., x→z and z→yx→z and z→y). For x,y∈V(T)x,y∈V(T), let b(x,y)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)b(x,y)>b(y,x) whenever y→xy→x. In this paper, a result shows that for a tournament on n players there exist exactly k   strong kings, 1⩽k⩽n1⩽k⩽n, with the following exceptions: k=n-1k=n-1 when n   is odd and k=nk=n when n is even. Moreover, we completely determine the uniqueness of tournaments.