### 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 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. 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 the 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 |
---|---|

Pages (from-to) | 168-181 |

Number of pages | 14 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 11 |

Issue number | 1 |

Publication status | Published - Feb 1998 |

### Fingerprint

### 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

*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.

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 11, no. 1, pp. 168-181.

}

TY - JOUR

T1 - Rankings of graphs

AU - Bodlaender, Hans L.

AU - Deogun, Jitender S.

AU - Jansen, Klaus

AU - Kloks, Ton

AU - Kratsch, Dieter

AU - Müller, Haiko

AU - Tuza, Z.

PY - 1998/2

Y1 - 1998/2

N2 - 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).

AB - 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).

KW - Edge ranking

KW - Graph algorithms

KW - Graph coloring

KW - Ranking of graphs

KW - Treewidth

KW - Vertex ranking

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UR - http://www.scopus.com/inward/citedby.url?scp=0041812815&partnerID=8YFLogxK

M3 - Article

VL - 11

SP - 168

EP - 181

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 1

ER -