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 ( 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.