An analytical study of the metric dimension in context of wheel related graphs

Abstract

An ordered set W=(w_1,…,w_k )⊆V(G) vertices of G is called a resolving set or locating set for G if every vertex is uniquely determined by its vector of distance 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 W_(n,m),Q and AWJ_2n, respectively. We prove that these infinite classes of convex polytopes generated by wheel, denoted by Q_n also gives a negative answer to an open problem proposed by Imran et al. (2012).