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. 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.