Document Type : Research Note

Authors

• M. Mohagheghy Nezhad ¹
• F. Rahbarnia ¹
• M. Mirzavaziri ²
• R. Ghanbari ¹

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.

Keywords

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