## 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 plane graphs has been determined in [3], [4], [5], [12], [14] and [18] while metric dimension of some families of convex polytopes has been studied in [8], [9], [10] and [11]and the following open problem was raised in [11]. Open Problem [11]: 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? In this paper, we extend this study to an infinite class of convex polytopes which is obtained as a combination of graph of an antiprism A _{n} [1] and graph of convex polytope Q _{n} [2] such that the outer cycle of A _{n} is the inner cycle of Q _{n}. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension. Note that the problem of determining whether (Hm(G) < k is an JVP-complete problem [7].

Original language | English |
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Pages (from-to) | 3-9 |

Number of pages | 7 |

Journal | Journal of Combinatorial Mathematics and Combinatorial Computing |

Volume | 81 |

Publication status | Published - May 2012 |

Externally published | Yes |

## Keywords

- Basis
- Convex polytope
- Metric dimension
- Plane graph
- Resolving set

## ASJC Scopus subject areas

- Mathematics(all)