Isomorphisms and traversability of directed path graphs

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.