On the metric dimension of few network sheets

Abstract

Let M = {v1, v2... vn} be an ordered set of vertices in a graph G (V, E). Then (d (u, v1), d (u, v2)...d (u, vn)) is called the M-coordinates of a vertex u of G. The set M is called a resolving set if the vertices of G have distinct M-coordinates. A metric basis is a resolving set M with minimum cardinality. If M is a metric basis then it is clear that for each pair of vertices u and v in the set of vertices V of G not in M, there is a vertex m in M such that the distance between u and m is not equal to the distance between v and m. The cardinality of a metric basis of G is called metric dimension. The members of a metric basis are called landmarks. A metric dimension problem is to find a metric basis. The problem of finding metric dimension is NP-Complete for general graphs. In this paper we have studied the metric dimension of a new graph called Octo-Nano windows, HDN like networks namely Equilateral Triangular Tetra sheets and Rectangular Tetra Sheet networks.