On the metric dimension of möbius ladders

If G is a connected graph, the distance d(u,v) between two vertices u,v ε V(G) is the length of a shortest path between them. Let W = {w ₁, ₂, ⋯,W _k} be an ordered set of vertices of C and let i; be a vertex of G. The representation γ(v; W) of v with respect to W is the κ-tuple {d{v ₁,w ₁),d{v,w ₂), d{v,W _k)). If distinct vertices of G have distinct representations with respect to W, then W is called a resolving set or locating set for G. A resolving set of minimum cardinality is called a basis for G and this cardi-nality is the metric dimension of G, denoted by dim(G). A family Q of connected graphs is a family with constant metric dimension if dim{G) does not depend upon the choice of G in Q. In this paper, we are dealing with the study of metric dimension of Möbius ladders. We prove that Möbius ladder M constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Möbius ladder Mn except when n = 2(mod 8). It is natural to ask for the charac-terization of regular graphs with constant metric dimension.