## Abstract

If G is a connected graph, the distance d(u,v) between two vertices u,v ε V(G) is the length of a shortest path between them. Let W = {w _{1},_{2}, ⋯,W_{k}} be an ordered set of vertices of C and let i; be a vertex of G. The representation γ(v; W) of v with respect to W is the κ-tuple {d{v_{1},w_{1}),d{v,w _{2}), d{v,W_{k})). If distinct vertices of G have distinct representations with respect to W, then W is called a resolving set or locating set for G. A resolving set of minimum cardinality is called a basis for G and this cardi-nality is the metric dimension of G, denoted by dim(G). A family Q of connected graphs is a family with constant metric dimension if dim{G) does not depend upon the choice of G in Q. In this paper, we are dealing with the study of metric dimension of Möbius ladders. We prove that Möbius ladder M constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Möbius ladder Mn except when n = 2(mod 8). It is natural to ask for the charac-terization of regular graphs with constant metric dimension.

Original language | English |
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Pages (from-to) | 403-410 |

Number of pages | 8 |

Journal | Ars Combinatoria |

Volume | 105 |

Publication status | Published - Jul 2012 |

Externally published | Yes |

## Keywords

- Basis
- Metric dimension
- Möbius ladder
- Resolving set

## ASJC Scopus subject areas

- General Mathematics