### Abstract

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.

Original language | English |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 24th International Workshop, WG 1998, Proceedings |

Editors | Juraj Hromkovic, Ondrej Sykora |

Publisher | Springer Verlag |

Pages | 114-123 |

Number of pages | 10 |

ISBN (Print) | 3540651950, 9783540651956 |

DOIs | |

Publication status | Published - 1998 |

Event | 24th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1998 - Smolenice Castle, Slovakia Duration: Jun 18 1998 → Jun 20 1998 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1517 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 24th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1998 |
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Country | Slovakia |

City | Smolenice Castle |

Period | 6/18/98 → 6/20/98 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Graph-Theoretic Concepts in Computer Science - 24th International Workshop, WG 1998, Proceedings*(pp. 114-123). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1517). Springer Verlag. https://doi.org/10.1007/10692760_10