On the existence of (k, l)-kernels in digraphs

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 (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) set of vertices. A k-kernel is a (k, k − 1)-kernel. For a strong digraph D, a set S ⊂ V (D) is a separator if D \ S is not strong, D is σ-strong if |V (D)| ≥ σ + 1 and has no separator with less than σ vertices. A digraph D is locally in(out)-semicomplete if whenever (v, u), (w, u) ∈ A(D) ((u, v), (u,w) ∈ A(D)), then (v, w) ∈ A(D) or (w, v) ∈ A(D). A digraph D is k-quasitransitive if the existence of a directed path (v0, v1, . . . , vk) in D implies that (v0, vk) ∈ A(D) or (vk, v0) ∈ A(D). In a digraph D which has at least one directed cycle, the length of a longest directed cycle is called its circumference. We propose the following conjecture, if D is a digraph with circumference l, then D has a l-kernel. This conjecture is proved for two families of digraphs and a partial result is obtained for a third family. In this article we prove that if D is a σ-strong digraph with circumference l, then D has a (k, (l− 1) + (l− σ) ⌊ k−2 σ ⌋ )-kernel for every k ≥ 2. Also, that if D is a locally in/out-semicomplete digraph such that, for a fixed integer l ≥ 1, (u, v) ∈ A(D) implies d(v, u) ≤ l, then D has a (k, l)-kernel for every k ≥ 2. As a consequence of this theorems we have that every (l−1)-strong digraph with circumference l and every locally out-semicomplete digraph with circumference l have an l-kernel, and every locally in-semicomplete digraph with circumference l has an l-solution. Also, we prove that every k-quasi-transitive digraph with circumference l ≤ k has an n-kernel for every n ≥ k.