## Abstract

An ordered set W={^{w1},...,^{wk}}V(G) of vertices of G is called a resolving set or locating set for G if every vertex is uniquely determined by its vector of distances to the vertices in W. A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension or location number of G, denoted by β(G). In this paper, we study the metric dimension of certain wheel related graphs, namely m-level wheels, an infinite class of convex polytopes and antiweb-gear graphs denoted by Wn, _{m},^{Qn} and AWJ2_{n}, respectively. We prove that these infinite classes of wheel related graphs have unbounded metric dimension. The study of an infinite class of convex polytopes generated by wheel, denoted by ^{Qn} also gives a negative answer to an open problem proposed by Imran et al. (2012) in [8]: Open Problem: Is it the case that the graph of every convex polytope has constant metric dimension? It is natural to ask for characterization of graphs with unbounded metric dimension.

Original language | English |
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Pages (from-to) | 624-632 |

Number of pages | 9 |

Journal | Applied Mathematics and Computation |

Volume | 242 |

DOIs | |

Publication status | Published - Sept 1 2014 |

Externally published | Yes |

## Keywords

- Basis
- Metric dimension
- Resolving set
- Wheel related graph

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics