On the existence of k-kernels in digraphs and in weighted 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 set of vertices (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) \N then exists v ∈ N such that d(u, v) ≤ l). A k-kernel is a (k, k − 1)-kernel. We propose an extension of the definition of (k, l)-kernel to (arc-)weighted digraphs, verifying which of the existing results for k-kernels are valid in this extension. If D is a digraph and w : A(D)→ Z is a weight function for the arcs of D, we can restate the problem of finding a k-kernel in the following way. If C is a walk in D, the weight of C is defined as w(C ) := ∑ f∈A(C) w(f). A subset S ⊆ V (D) is (k, w)-independent if, for every u, v ∈ S there does not exist an uv-directed path of weight less than k. A subset S ⊆ V (D) will be (l, w)-absorbent if, for every u ∈ V (D) \S, there exists an uS-directed path of weight less than or equal to l. A subset N ⊆ V (D) is a (k, l, w)-kernel if it is (k, w)-independent and (l, w)-absorbent. We prove, among other results, that every transitive digraph has a (k, k−1, w)-kernel for every k, that if T is a tournament and w(a) ≤ k−1 2 for every a ∈ A(T ), then T has a (k, w)-kernel and that if every directed cycle in a quasi-transitive digraph D has weight ≤ k−1 2 + 1, then D has a (k, w)-kernel. Also, we let the weight function to have an arbitrary group as codomain and propose another variation of the concept of k-kernel.