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aequationes mathematicae

, Volume 59, Issue 1–2, pp 45–54 | Cite as

The partition dimension of a graph

  • G. Chartrand
  • E. Salehi
  • P. Zhang

Summary.

For a vertex v of a connected graph G and a subset S of V(G), the distance between v and S is \( d(v, S) = \min \{d(v, x) | x \in S\} \). For an ordered k-partition \( \Pi = \{S_1, S_2, \cdots, S_k\} \)> of V(G), the representation of v with respect to \( \Pi \) is the k-vector \( r(v | \Pi) = (d(v, S_1), \,d(v, S_2), \cdots, \,d(v, S_k)) \). The k-partition \( \Pi \) is a resolving partition if the k-vectors \( r(v | \Pi), \,v \in V(G) \), are distinct. The minimum k for which there is a resolving k-partition of V(G) is the partition dimension pd(G) of G. It is shown that the partition dimension of a graph G is bounded above by 1 more than its metric dimension. An upper bound for the partition dimension of a bipartite graph G is given in terms of the cardinalities of its partite sets, and it is shown that the bound is attained if and only if G is a complete bipartite graph. Graphs of order n having partition dimension 2, n, or n— 1 are characterized.

Keywords

Bipartite Graph Connected Graph Complete Bipartite Graph Partition Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag, Basel, 2000

Authors and Affiliations

  • G. Chartrand
    • 1
  • E. Salehi
    • 1
  • P. Zhang
    • 2
  1. 1.Department of Mathematics and Statistics, West Michigan University, Kalamazoo, MI 49008, USA US
  2. 2.Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, NV 89154, USA US

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