An infinite class 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 [3], [4], [5], [12], [14] and [18] while metric dimension of some families of convex polytopes has been studied in [8], [9], [10] and [11]and the following open problem was raised in [11]. Open Problem [11]: Let G be the graph of a convex polytope which is obtained by joining the graph of two different convex polytopes G ₁ and G ₂ (such that the outer cycle of G ₁ is the inner cycle of G ₂) both having constant metric dimension. Is it the case that G will always have the constant metric dimension? In this paper, we extend this study to an infinite class of convex polytopes which is obtained as a combination of graph of an antiprism A _n [1] and graph of convex polytope Q _n [2] such that the outer cycle of A _n is the inner cycle of Q _n. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension. Note that the problem of determining whether (Hm(G) < k is an JVP-complete problem [7].