Rankings of graphs

  • H. L. Bodlaender
  • J. S. Deogun
  • K. Jansen
  • T. Kloks
  • D. Kratsch
  • H. Müller
  • Zs. Tuza
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 903)


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\(\chi '_r \left( G \right)\)) 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 \(\chi '_r \left( {K_n } \right)\).


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Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • H. L. Bodlaender
    • 1
  • J. S. Deogun
    • 2
  • K. Jansen
    • 3
  • T. Kloks
    • 4
  • D. Kratsch
    • 5
  • H. Müller
    • 5
  • Zs. Tuza
    • 6
  1. 1.Department of Computer ScienceUtrecht UniversityTB UtrechtThe Netherlands
  2. 2.Department of Computer Science and EngineeringUniversity of Nebraska-LincolnLincolnUSA
  3. 3.Fachbereich IV, Mathematik und InformatikUniversität TrierTrierGermany
  4. 4.Department of Mathematics and Computing ScienceEindhoven University of TechnologyMB EindhovenThe Netherlands
  5. 5.Fakultät für Mathematik und InformatikFriedrich-Schiller-UniversitätJenaGermany
  6. 6.Computer and Automation InstituteHungarian Academy of SciencesBudapestHungary

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