Computing the metric dimension of wheel related graphs

An ordered set 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). 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,^Qn and AWJ2_n, 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 ^Qn 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.