### Abstract

In this paper, we define distance partition of vertex set of a graph G with reference to a vertex in it and with the help of the same, a graph with metric dimension two (i.e. β (G) = 2) is characterized. In the process, we develop a polynomial time algorithm that verifies if the metric dimension of a given graph G is two. The same algorithm explores all metric bases of graph G whenever β (G) = 2. We also find a bound for cardinality of any distance partite set with reference to a given vertex, when ever β (G) = 2. Also, in a graph G with β (G) = 2, a bound for cardinality of any distance partite set as well as a bound for number of vertices in any sub graph H of G is obtained in terms of diam H.

Original language | English |
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Pages (from-to) | 622-627 |

Number of pages | 6 |

Journal | World Academy of Science, Engineering and Technology |

Volume | 36 |

Publication status | Published - 01-12-2009 |

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### All Science Journal Classification (ASJC) codes

- Engineering(all)

### Cite this

*World Academy of Science, Engineering and Technology*,

*36*, 622-627.