aequationes mathematicae

, Volume 59, Issue 1–2, pp 45–54

# 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

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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