Rankings of graphs

H. L. Bodlaender, J. S. Deogun, K. Jansen, T. Kloks, D. Kratsch, H. Müller, Zs Tuza

Research output: Chapter in Book/Report/Conference proceedingConference contribution

19 Citations (Scopus)

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 χr (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 χr (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 χr and the chromatic number χ coincide on all induced subgraphs, show that χr (G)=χ(G) implies χ(G)=ω(G) (largest clique size) and give a formula for χ′r(Kn).

Original languageEnglish
Title of host publicationGraph-Theoretic Concepts in Computer Science - 20th International Workshop, WG 1994, Proceedings
EditorsErnst W. Mayr, Gunther Schmidt, Gottfried Tinhofer
PublisherSpringer Verlag
Pages292-304
Number of pages13
ISBN (Print)3540590714, 9783540590712
Publication statusPublished - Jan 1 1995
Event20th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1994 - Herrsching, Germany
Duration: Jun 16 1994Jun 18 1994

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume903
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other20th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 1994
CountryGermany
CityHerrsching
Period6/16/946/18/94

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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  • Cite this

    Bodlaender, H. L., Deogun, J. S., Jansen, K., Kloks, T., Kratsch, D., Müller, H., & Tuza, Z. (1995). Rankings of graphs. In E. W. Mayr, G. Schmidt, & G. Tinhofer (Eds.), 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.