Ranking numbers of graphs

  • Chang C
  • Kuo D
  • Lin H
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Abstract

Given a graph G, a vertex ranking (or simply, ranking) of G is a mapping f from V (G) to the set of all positive integers, such that for any path between two distinct vertices u and v with f (u) = f (v), there is a vertex w in the path with f (w) > f (u). If f is a ranking of G, the ranking number of G under f, denoted γf (G), is defined by γf (G) = max {f (v) : v ∈ V (G)}, and the ranking number of G, denoted γ (G), is defined by γ (G) = min {γf (G) : f is a ranking of G}. The vertex ranking problem is to determine the ranking number γ (G) of a given graph G. This problem is a natural model for the manufacturing scheduling problem. We study the ranking numbers of graphs in this paper. We consider the relation between the ranking numbers and the minimal cut sets, and the relation between the ranking numbers and the independent sets. From this, we obtain the ranking numbers of the powers of paths and the powers of cycles, the Cartesian product of P2 with Pn or Cn, and the caterpilars. And we also find the vertex ranking numbers of the composition of two graphs in this paper. © 2010 Elsevier B.V. All rights reserved.

Author-supplied keywords

  • Cartesian product
  • Caterpillar
  • Combinatorial problems
  • Composition
  • Cycle
  • Independent set
  • Minimal cut set
  • Path
  • Power
  • Ranking number
  • Vertex ranking

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Authors

  • Chan Wei Chang

  • David Kuo

  • Hsing Ching Lin

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