### k-kernels in k-transitive and k-quasi-transitive digraphs

César Hernández-Cruz, Hortensia Galeana-Sánchez
2012 Discrete Mathematics
k, l)-kernel k-kernel Transitive digraph Quasi-transitive digraph a b s t r a c t Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. 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
more » ... hs, two of them generalizing transitive and quasi-transitive digraphs respectively; a digraph D is k-transitive if whenever (x 0 , x 1 , . . . , x k ) is a path of length k in D, then (x 0 , x k ) ∈ 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.