On the existence and number of -kings in -quasi-transitive digraphs

Let D = ( V ( D ) , A ( D ) ) be a digraph and k ≥ 2 an integer. We say that D is k -quasi-transitive if for every directed path ( v 0 , v 1 , … , v k ) in D we have ( v 0 , v k ) ∈ A ( D ) or ( v k , v 0 ) ∈ A ( D ) . Clearly, a 2 -quasi-transitive digraph is a quasi-transitive digraph in the usual sense. ensen and Gutin proved that a quasi-transitive digraph D has a 3 -king if and only if D has a unique initial strong component, and if D has a 3 -king and the unique initial strong component of D has at least three vertices, then D has at least three 3 -kings. In this paper we prove the following generalization: a k -quasi-transitive digraph D has a ( k + 1 ) -king if and only if D has a unique initial strong component, and if D has a ( k + 1 ) -king, then either all the vertices of the unique initial strong components are ( k + 1 ) -kings or the number of ( k + 1 ) -kings in D is at least ( k + 2 ) . Also, we obtain new results on the minimum number of 3 -kings in quasi-transitive digraphs.