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 infinite outward path for i?{1,2}, then D has a kernel.

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