Computing the metric dimension of wheel related graphs

An ordered set W={w1,…,wk}⊆V(G)W={w1,…,wk}⊆V(G) of vertices of G is called a resolving set or locating set for G if every vertex is uniquely determined by its vector of distances to the vertices in W. A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension or location number   of G, denoted by β(G)β(G).In this paper, we study the metric dimension of certain wheel related graphs, namely m  -level wheels, an infinite class of convex polytopes and antiweb-gear graphs denoted by Wn,m,QnWn,m,Qn and AWJ2nAWJ2n, respectively. We prove that these infinite classes of wheel related graphs have unbounded metric dimension. The study of an infinite class of convex polytopes generated by wheel, denoted by QnQn also gives a negative answer to an open problem proposed by Imran et al. (2012) in [8]:Open Problem: Is it the case that the graph of every convex polytope has constant metric dimension?It is natural to ask for characterization of graphs with unbounded metric dimension.