k-kernels in multipartite tournaments

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent set of vertices (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l). A k-kernel is a (k, k − 1)-kernel. An m-partite tournament is an orientation of an m-partite complete graph. In this paper we introduce a new tool for finding sufficient conditions for a digraph to have a k-kernel, the ktransitive closure of a digraph, which is used to prove that every m-partite tournament has a k-kernel for every k ≥ 4, m ≥ 2, and to give a simple proof of the fact that for k ≥ 2 is NP -complete to decide if a digraph D has a k-kernel, or not. We also characterize the m-partite tournaments that have a 3-kernel for m ≥ 2 as those m-partite tournaments having a 2-absorbent vertex for the directed cycles of length four. keywords: digraph, kernel, (k, l)-kernel, k-kernel, tournament, multipartite tournament. AMS Subject Classification: 05C20.