Graphs with Metric Dimension Two-A Characterization

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. 2 ) ( = G β ) 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 2 ) ( = G β . We also find a bound for cardinality of any distance partite set with reference to a given vertex, when ever 2 ) ( = G β . Also, in a graph G with 2 ) ( = G β , 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 . Keywords—Metric basis, Distance partition, Metric dimension.