quasi - transitive digraphs ∗

Let D = (V , A) be a directed graph (digraph) without loops nor multiple arcs. A set of vertices S of a digraph D is a (k, l)-kernel of D if and only if for any two vertices u, v in S, d(u, v) ≥ k and for any vertex u in V \ S there exists v in S such that d(u, v) ≤ l. A digraph D is called quasi-transitive if and only if for any distinct vertices u, v, w of D such that u→ v → w, then u and w are adjacent vertices in D. In this paper, we characterize the (2, 1)-kernels (usually knowns as kernels simply) in quasi-transitive digraphs. We prove also that every quasi-transitive digraph possesses a (3, 2)-kernel and we give how to get it. 2000 Mathematic Subject Classification: Primary: 05C20; Secondary: 05C69.