Kernels in pretransitive digraphs

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A kernel N of D is an independent set of vertices such that for every w∈V (D) − N there exists an arc from w to N . A digraph D is called right-pretransitive (resp. left-pretransitive) when (u; v)∈A(D) and (v; w)∈A(D) implies (u; w)∈A(D) or (w; v)∈A(D) (resp. (u; v)∈A(D) and (v; w)∈A(D) implies (u; w)∈A(D) or (v; u)∈A(D)). This concepts were introduced by P. Duchet in 1980. In this paper is proved the following result: Let D be a digraph. If D=D1∪D2 where D1 is a right-pretransitive digraph, D2 is a left-pretransitive digraph and Di contains no in=nite outward path for i∈{1; 2}, then D has a kernel. c © 2003 Elsevier B.V. All rights reserved.