Rankings of Directed Graphs

(Extended abstract)
  • Jan Kratochvíl
  • Zsolt Tuza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)


A ranking of a graph is a coloring of the vertex set with positive integers such that on every path connecting two vertices of the same color there is a vertex of larger color. We consider the directed variant of this problem, where the above condition is imposed only on those paths in which all edges are oriented in the same direction. We show that the ranking number of a directed tree is bounded by that of its longest directed path plus one, and that it can be computed in polynomial time. Unlike the undirected case, however, deciding whether the ranking number of a directed (and even of an acyclic directed) graph is bounded by a constant is NP-complete. In fact, the 3-ranking of planar bipartite acyclic digraphs is already hard.


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  1. 1.
    Bodlaender, H., Deogun, J.S., Jansen, K., Kloks, T., Kratsch, D., Müller, H., Tuza, Z.: Rankings of graphs. In: Mayr, E.W., et al. (eds.) WG 1994. LNCS, vol. 903, pp. 292–304. Springer, Heidelberg (1995); Extended version in: SIAM J. Discr. Math. 11, 168–181 (1998)CrossRefGoogle Scholar
  2. 2.
    Hujter, M., Tuza, Z.: Precoloring extension. II. Graph classes related to bipartite graphs. Acta Math. Univ. Carolinae 62, 1–11 (1993)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Hujter, M., Tuza, Z.: Precoloring extension. III. Classes of perfect graphs. Combin. Probab. Computing 5, 35–56 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kratochvíl, J.: Precoloring extension with fixed color bound. Acta Math. Univ. Carolinae 62, 139–153 (1993)Google Scholar
  5. 5.
    Kratochvíl, J., Sebö, A.: Coloring precolored perfect graphs. J. Graph Theory 25, 207–215 (1995)Google Scholar
  6. 6.
    Lam, T.W., Yue, F.L.: Egde ranking is NP-complete. To appear in Discrete Applied Math.Google Scholar
  7. 7.
    Tuza, Z.: Graph colorings with local constraints—A survey. Discuss. Math. Graph Theory 17, 161–228 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Jan Kratochvíl
    • 1
  • Zsolt Tuza
    • 2
  1. 1.Department of Applied MathematicsCharles UniversityPraha 1Czech Republic
  2. 2.Computer and Automation InstituteHungarian Academy of SciencesBudapestHungary

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