Geodesic
Isomorphisms and traversability of directed path graphs
Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 2, pp. 215-228

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The concept of a line digraph is generalized to that of a directed path graph. The directed path graph Pₖ(D) of a digraph D is obtained by representing the directed paths on k vertices of D by vertices. Two vertices are joined by an arc whenever the corresponding directed paths in D form a directed path on k+1 vertices or form a directed cycle on k vertices in D. In this introductory paper several properties of P₃(D) are studied, in particular with respect to isomorphism and traversability. In our main results, we characterize all digraphs D with P₃(D) ≅ D, we show that P₃(D₁) ≅ P₃(D₂) "almost always" implies D₁ ≅ D₂, and we characterize all digraphs with Eulerian or Hamiltonian P₃-graphs.
Keywords: directed path graph, line digraph, isomorphism, travers-ability 
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Broersma, Hajo; Li, Xueliang. Isomorphisms and traversability of directed path graphs. Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 2, pp. 215-228. http://geodesic.mathdoc.fr/item/DMGT_2002_22_2_a0/