Shahrood University of TechnologyJournal of Algebraic Systems2345-51287220200101A CHARACTERIZATION FOR METRIC TWO-DIMENSIONAL GRAPHS AND THEIR ENUMERATION179187158810.22044/jas.2019.7367.1363ENM. Mohagheghy NezhadDepartment of Applied Mathematics, Ferdowsi University of Mashhad, P.O. Box
1159, Mashhad, Iran.F. RahbarniaDepartment of Applied Mathematics, Ferdowsi University of Mashhad, P.O. Box
1159, Mashhad, Iran.M. MirzavaziriDepartment of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159,
Mashhad, Iran.R. GhanbariDepartment of Applied Mathematics, Ferdowsi University of Mashhad, P.O. Box
1159, Mashhad, Iran.Journal Article20180815The \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 $B$. <br /> 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.https://jas.shahroodut.ac.ir/article_1588_14ce71a7aec0d0417b21b3acf6be72d4.pdf