Rankings of directed graphs

Jan Kratochvíl, Zsolt Tuza

Research output: Contribution to journalArticle

2 Citations (Scopus)


A ranking of a graph is a coloring of the vertex set with positive integers in such a way 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 consecutively. We show that the ranking number of an orientation of a 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.

Original languageEnglish
Pages (from-to)374-384
Number of pages11
JournalSIAM Journal on Discrete Mathematics
Issue number3
Publication statusPublished - Sep 1999


  • Algorithm
  • Directed graph
  • Graph
  • NP-completeness
  • Oriented graph
  • Ranking

ASJC Scopus subject areas

  • Mathematics(all)

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