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 convex polytopes has been determined in [8] and an open problem was raised: Is it the case that the graph of every convex polytope has constant metric dimension? A fundamental question in graph theory concerns how the value of a parameter is affected by making a small change in the graph? In this paper,by giving answer to open problem proposed in [8],we study the metric dimension of some classes of convex polytopes which are obtained from the graph of convex polytopes defined in [1] and [2] 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. Also,we see that the metric dimension of these classes of convex polytopes is same as the graph of convex polytopes defined in [1] and [2],so we raise a question in more general form regarding the metric dimension of new classes of convex polytopes obtained from old ones by adding new edges and having the same vertex set. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.