Kernels in quasi-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 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 is called quasi-transitive when ( u , v ) ∈ A ( D ) and ( v , w ) ∈ A ( D ) implies ( u , w ) ∈ A ( D ) or ( w , u ) ∈ A ( D ) . This concept was introduced by Ghouilá–Houri [Caractérisation des graphes non orientés dont on peut orienter les arrêtes de maniere à obtenir le graphe d’ un relation d’ordre, C.R. Acad. Sci. Paris 254 (1962) 1370–1371] and has been studied by several authors. In this paper the following result is proved: Let D be a digraph. Suppose D = D 1 ∪ D 2 where D i is a quasi-transitive digraph which contains no asymmetrical infinite outward path (in D i ) for i ∈ { 1 , 2 } ; and that every directed cycle of length 3 contained in D has at least two symmetrical arcs, then D has a kernel. All the conditions for the theorem are tight.