On metric dimension of convex polytopes with pendant edges

A family G of connected graphs is said to be a family with constant metric dimension if dim(G) does not depend upon the choice of G in G. In this paper we study the metric dimension of some plane graphs which are obtained from some convex polytopes by attaching a pendant edge to each vertex of the outer cycle in a plane representation of these convex polytopes. We prove that the metric dimension of these plane graphs is constant and only three vertices appropriately chosen suffice to resolve all the vertices of these classes of graphs. It is natural to ask for the characterization of graphs G which are plane representations of convex polytopes having the property that dim(G) = dim(G'), where G' is obtained from G by attaching a pendant edge to each vertex of the outer cycle of G.