## Abstract

A family G of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in g The metric dimension of some classes of convex polytopes has been determined in [8-12] and an open problem was raised in [10]: Let G be the graph of a convex polytope which is obtained by joining the graph of two different convex polytopes G _{1} and G_{2} (such that the outer cycle of G_{1} is the inner cycle of G_{2}) both having constant metric dimension. Is it the case that G will always have the constant metric dimension?this paper, we study the metric dimension of an infinite classes of convex polytopes which are obtained by the combinations of two different graph of convex polytopes. It is shown that this infinite class of convex polytoes has constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes.

Original language | English |
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Pages (from-to) | 197-205 |

Number of pages | 9 |

Journal | Journal of Combinatorial Mathematics and Combinatorial Computing |

Volume | 77 |

Publication status | Published - May 2011 |

Externally published | Yes |

## Keywords

- Antiprism
- Basis
- Convex polytopes
- Metric dimension
- Planar graph
- Prsism
- Resolving set

## ASJC Scopus subject areas

- Mathematics(all)