k-Kings in k-Quasitransitive Digraphs

Let D be a digraph with vertex set VD and arc set AD. A vertex x is a k-king of D, if for every y∈VD, there is an x,y-path of length at most k. A subset N of VD is k-independent if for every pair of vertices u,v∈N, we have du,vi¾?k and dv,ui¾?k; it is l-absorbent if for every u∈VD-N there exists v∈N such that du,vi¾?l. A k,l-kernel of D is a k-independent and l-absorbent subset of VD. A k-kernel is a k,k-1-kernel. A digraph D is k-quasitransitive, if for any path x0x1...xk of length k, x0 and xk are adjacent. In this article, we will prove that a k-quasitransitive digraph with ki¾?4 has a k-king if and only if it has a unique initial strong component and the unique initial strong component is not isomorphic to an extended k+1-cycle C[E0,E1,...,Ek] where each Ei has at least two vertices. Using this fact, we show that every strong k-quasitransitive digraph has a k+1-kernel.