Geodesic
3-transitive digraphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 2, pp. 205-219

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Let D be a digraph, V(D) and A(D) will denote the sets of vertices and arcs of D, respectively. A digraph D is 3-transitive if the existence of the directed path (u,v,w,x) of length 3 in D implies the existence of the arc (u,x) ∈ A(D). In this article strong 3-transitive digraphs are characterized and the structure of non-strong 3-transitive digraphs is described. The results are used, e.g., to characterize 3-transitive digraphs that are transitive and to characterize 3-transitive digraphs with a kernel.
Keywords: digraph, kernel, transitive digraph, quasi-transitive digraph, 3-transitive digraph, 3-quasi-transitive digraph 
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     author = {Hern\'andez-Cruz, C\'esar},
     title = {3-transitive digraphs},
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Hernández-Cruz, César. 3-transitive digraphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 2, pp. 205-219. http://geodesic.mathdoc.fr/item/DMGT_2012_32_2_a1/