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

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.

Keywords