Abstract: In this paper, we
define distance partition of vertex set of a graph Gwith reference to a vertex in it and with the help of the same, a graph
with metric dimension two is characterized. In the process, we
develop a polynomial time algorithm that verifies if the metric dimension of a
given graph Gis two. The
same algorithm explores all metric bases of graph Gwhenever We also find a bound for cardinality
of any distance partite set with reference to a given vertex, whenever Also, in a graph Gwith a bound for cardinality of any
distance partite set as well as a bound for number of vertices in any subgraph Hof Gis obtained
in terms of diam H.
Keywords and phrases: metric dimension, metric basis, distance partition.
