A CHARACTERIZATION FOR METRIC TWO-DIMENSIONAL GRAPHS AND THEIR ENUMERATION

Document Type: Research Note

Authors

1 Department of Applied Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad, Iran.

2 Department of Pure Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad, Iran.

Abstract

‎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.

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