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 subset N of V(D) is k-independent if for every pair of vertices u,v∈N, we have d(u,v),d(v,u)≥k; it is l-absorbent if for every u∈V(D)−N there exists v∈N such that d(u,v)≤l. A (k,l)-kernel of D is a k-independent and l-absorbent subset of V(D). A k-kernel is a (k,k−1)-kernel.A digraph D is transitive if for every path (u,v,w) in D we have (u,w)∈A(D). This concept can be generalized as follows, a digraph D is quasi-transitive if for every path (u,v,w) in D, we have (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.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 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; 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.