### Abstract

A vertex (edge) coloring c∶V → {1, 2, ⋯, t} (c′∶E → {1, 2, ⋯, t}) of a graph G=(V, E) is a vertex (edge) t-ranking if for any two vertices (edges) of the same color every path between them contains a vertex (edge) of larger color. The vertex ranking number χ_{r} (G) (edge ranking number χ′_{r}(G)) is the smallest value of t such that G has a vertex (edge) t-ranking. In this paper we study the algorithmic complexity of the VERTEX RANKING and EDGE RANKING problems. Among others it is shown that χ_{r} (G) can be computed in polynomial time when restricted to graphs with treewidth at most k for any fixed k. We characterize those graphs where the vertex ranking number χ_{r} and the chromatic number χ coincide on all induced subgraphs, show that χ_{r} (G)=χ(G) implies χ(G)=ω(G) (largest clique size) and give a formula for χ′_{r}(K_{n}).

Original language | English (US) |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 20th International Workshop, WG 1994, Proceedings |

Editors | Ernst W. Mayr, Gunther Schmidt, Gottfried Tinhofer |

Publisher | Springer Verlag |

Pages | 292-304 |

Number of pages | 13 |

ISBN (Print) | 3540590714, 9783540590712 |

Publication status | Published - Jan 1 1995 |

Event | 20th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1994 - Herrsching, Germany Duration: Jun 16 1994 → Jun 18 1994 |

### 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 | 903 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 20th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1994 |
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Country | Germany |

City | Herrsching |

Period | 6/16/94 → 6/18/94 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Graph-Theoretic Concepts in Computer Science - 20th International Workshop, WG 1994, Proceedings*(pp. 292-304). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 903). Springer Verlag.