Computing Metric Dimension of M-Level Wheel Related Graphs

Abstract

An ordered subset of nodes of is called a resolving set or locating set for if each node is individually determined by its code of distances to the nodes in . A resolving set of minimum number of nodes is called a basis for and this cardinality is the metric dimension or location number of , represented as In this thesis, we compute the metric dimension of some wheel related graphs, such as m-level gear graph, m-level antiweb-wheel graph, m-level antiweb-gear graph and an infinite class of convex polytopes denoted by , respectively. We prove that the metric dimension of aforementioned wheel related graphs and a family of convex polytope is not bounded. The unbounded metric dimension of convex polytope also furnish a negative answer to the problem given in [19]: 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.