K-kernels in K-transitive and K-quasi-transitive 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. A digraph D is transitive if (u, v), (v, w) ∈ A(D) implies that (u,w) ∈ A(D). This concept can be generalized as follows, a digraph D is quasi-transitive if (u, v), (v, w) ∈ A(D) implies that (u,w) ∈ A(D) or (w, u) ∈ A(D). In the literature beautiful results describing the structure of both transitive and quasi-transitive digraphs are found that can be used to prove that every transitive digraph has a k-kernel for every k ≥ 2 and that every quasi-transitive digraph has a k-kernel for every k ≥ 3. Let us recall that is NP -complete to decide if a digraph D has a k-kernel. We introduce three new families of digraphs, two of them generalizing transitive and quasi-transitive digraphs respectively; a digraph D is k-transitive if whenever (x0, x1, . . . , xk) is a directed path of length k in D, then (x0, xk) ∈ A(D); k-quasi-transitive digraphs are analogously defined, so (quasi)transitive digraphs are 2-(quasi-)transitive digraphs. We prove some structural results about both classes of digraphs that can be used to prove that a k-transitive digraph has an n-kernel for every n ≥ k; that for even k ≥ 2, every k-quasi-transitive digraph has an n-kernel for every n ≥ k + 2; and that every 3-quasi-transitive digraph has k-kernel for every k ≥ 4. Also, we prove that a k-transitive digraph has a k-king if and only if it has a unique initial strong component and that a k-quasi-transitive digraph has a (k + 1)-king if and only if it has a unique initial strong component. keywords: digraph, kernel, (k, l)-kernel, k-kernel, transitive digraph, quasi-transitive digraph AMS Subject Classification: 05C20.