## 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## Preview

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