César Hernández-Cruz

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.

k-kernels in generalizations of 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 set of vertices (if u, v ∈ N , u 6= v, 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). A k-kernel is a (k, k − 1)-kernel. Quasi-transitive, right-pretransitive and left-pretransitive digraphs are generalizations of transitive digraphs. In this paper the following results are proved: Let D be a right-(left-) pretransitive strong digraph such that every directed triangle of D is symmetrical, then D has a k-kernel for every integer k ≥ 3; the result is also valid for non-strong digraphs in the right-pretransitive case. We also give a proof of the fact that every quasi-transitive digraph has a (k, l)-kernel for every integers k > l ≥ 3 or k = 3 and l = 2. keywords: digraph, kernel, (k, l)-kernel, k-kernel, transitive digraph, quasi-transitive digraph, rightpretransitive digraph, left-pretransitive digraph, pretransitive digraph. AMS Subject Classification: 05C20.

3-transitive digraphs

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is 3-transitive if the existence of the directed path (u, v, w, x) of length 3 in D implies the existence of the arc (u, x) ∈ A(D). In this article strong 3-transitive digraphs are characterized and the structure of non-strong 3-transitive digraphs is described. The results are used, e.g, to characterize 3-transitive digraphs that are transitive and to characterize 3-transitive digraphs with a kernel. keywords: digraph, transitive digraph, quasi-transitive digraph, 3-transitive digraph, 3-quasi-transitive digraph, kernel. AMS Subject Classification: 05C20.

On the existence and number of (k+1)-kings in k-quasi-transitive digraphs

Let

On the Existence of (k,l)-Kernels in Infinite Digraphs: A Survey

D=(V(D), A(D))

On the Complexity of the 3-Kernel Problem in Some Classes of Digraphs

be a digraph and

Cyclically k-partite digraphs and k-kernels

k \ge 2

4-Transitive Digraphs I: The Structure of Strong 4-Transitive Digraphs

an integer. We say that

Some Remarks On The Structure Of Strong K-Transitive Digraphs

D

Strict chordal and strict split digraphs

is