M. Mohagheghy Nezhad - Department of Applied Mathematics, Ferdowsi University of Mashhad, P.O. Box ۱۱۵۹, Mashhad, Iran.
F. Rahbarnia - Department of Applied Mathematics, Ferdowsi University of Mashhad, P.O. Box ۱۱۵۹, Mashhad, Iran.
M. Mirzavaziri - Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box ۱۱۵۹, Mashhad, Iran.
R. Ghanbari - Department of Applied Mathematics, Ferdowsi University of Mashhad, P.O. Box ۱۱۵۹, Mashhad, Iran.

‎The textit{metric dimension} of a connected graph $G$ is the minimum number of vertices in a subset $B$ of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $B$‎. ‎In this case‎, ‎$B$ is called a textit{metric basis} for $G$‎. ‎The textit{basic distance} of a metric two dimensional graph $G$ is the distance between the elements of $B$‎. ‎Giving a characterization for those graphs whose metric dimensions are two‎, ‎we enumerate the number of $n$ vertex metric two dimensional graphs with basic distance 1‎.

Metric dimension, Resolving set, Metric basis, Basic distance, Contour of a graph