New classes of convex polytopes with constant metric dimension

A family G of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G. The metric dimension of some classes of plane graphs has been determined in [2], [3], [4], [12], [15] and [22] while metric dimension of some families of convex polytopes has been studied in [9], [10] and [11] and the following open problem was raised in [10]. Open Problem [10]: Let G' be the graph of convex polytope obtained from the graph of convex polytope G by adding new edges in G such that V(G') = V(G). Is it the case that G' and G will always have the same metric dimension? In this paper, we extend this study by considering some classes of convex polytopes which are obtained from the graph of convex polytope S_n defined in [11] by adding new edges in it and having the same vertex set. It is shown that these classes of convex polytoes have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. A conjecture in more general meaning is also proposed in this regard. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.