Rankings of graphs

Hans L. Bodlaender, Jitender S. Deogun, Klaus Jansen, Ton Kloks, Dieter Kratsch, Haiko Müller, Z. Tuza

Research output: Contribution to journalArticle

90 Citations (Scopus)

Abstract

A vertex (edge) coloring Φ : V → {1, 2,...,t} (Φ′ : 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. 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 the graphs where the vertex ranking number χrand 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
Pages (from-to)168-181
Number of pages14
JournalSIAM Journal on Discrete Mathematics
Volume11
Issue number1
Publication statusPublished - Feb 1998

Fingerprint

Ranking
Color
Coloring
Graph in graph theory
Vertex of a graph
Polynomials
Algorithmic Complexity
Vertex Coloring
Treewidth
Edge Coloring
Induced Subgraph
Chromatic number
Clique
Polynomial time
Imply
Path

Keywords

  • Edge ranking
  • Graph algorithms
  • Graph coloring
  • Ranking of graphs
  • Treewidth
  • Vertex ranking

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

Bodlaender, H. L., Deogun, J. S., Jansen, K., Kloks, T., Kratsch, D., Müller, H., & Tuza, Z. (1998). Rankings of graphs. SIAM Journal on Discrete Mathematics, 11(1), 168-181.

Rankings of graphs. / Bodlaender, Hans L.; Deogun, Jitender S.; Jansen, Klaus; Kloks, Ton; Kratsch, Dieter; Müller, Haiko; Tuza, Z.

In: SIAM Journal on Discrete Mathematics, Vol. 11, No. 1, 02.1998, p. 168-181.

Research output: Contribution to journalArticle

Bodlaender, HL, Deogun, JS, Jansen, K, Kloks, T, Kratsch, D, Müller, H & Tuza, Z 1998, 'Rankings of graphs', SIAM Journal on Discrete Mathematics, vol. 11, no. 1, pp. 168-181.
Bodlaender HL, Deogun JS, Jansen K, Kloks T, Kratsch D, Müller H et al. Rankings of graphs. SIAM Journal on Discrete Mathematics. 1998 Feb;11(1):168-181.
Bodlaender, Hans L. ; Deogun, Jitender S. ; Jansen, Klaus ; Kloks, Ton ; Kratsch, Dieter ; Müller, Haiko ; Tuza, Z. / Rankings of graphs. In: SIAM Journal on Discrete Mathematics. 1998 ; Vol. 11, No. 1. pp. 168-181.
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