k-kernels in generalizations of transitive digraphs

Abstract

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 = 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.

k‐Kings in k‐Quasitransitive Digraphs

“…Let D be a k ‐quasitransitive digraph. Then D has a k ‐serf if and only if it has an unique terminal strong component and the unique terminal strong component is not isomorphic to an extended ‐cycle where each has at least two vertices. Theorem Let D be a quasitransitive digraph. Then D has a k ‐kernel for every integer . Theorem Let D be a 3‐quasitransitive digraph.…”

“…Theorem 2.10. [4] Let D be a quasitransitive digraph. Then D has a k-kernel for every integer k ≥ 3.…”

(k+1)-kernels and the number ofk-kings inWang1 2015Discrete Mathematics

“…This is the best possible result in terms of the number of k-kings in a k-quasi-transitive digraph. Let the digraph 6 , y 6 }. A semicomplete bipartite digraph must be a 3-quasi-transitive digraph.…”

“…In Ref. [6], GaleanaSánchez and Hernández-Cruz studied the existence of k-kernels in quasi-transitive digraphs. In Ref.…”

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

“…, v k+1 ), of length k + 1, with two additional arcs, (v k , v 0 ) and (v k+1 , v 1 ). It is direct to observe that D is a strong k-quasi-transitive digraph without a k-king, nonetheless, {v 0 , v k+1 } is a (k + 1)-kernel of D. It was proved in [7] that every 2-quasitransitive digraph has a 3-kernel and in [6] that every 3-quasi-transitive digraph has a 4-kernel. So the following conjecture also stated in [6] seems reasonable.…”