Resolvability and fault-tolerant resolvability structures of convex polytopes

Abstract

In this paper, we study resolvability and fault-tolerant resolvability of convex polytopes and related geometric graphs. Imran et al. (2010) [18] raised an open problem asserting that whether or not every family of convex polytope has a constant metric dimension. Raza et al. (2018) [28] studied the fault-tolerant metric dimension of certain families of convex polytopes. They also concluded their study with an open problem asking whether or not every family of convex polytope has a constant fault-tolerant metric dimension. In this paper, we provide negative answers to both of the aforementioned open problems. We answer the first question by constructing a family of convex polytopes with an unbounded metric dimension. By proving a result between resolvability and fault-tolerant resolvability structures of a graph, we show that the family of convex polytope with an unbounded metric dimension also possesses an unbounded fault-tolerant resolvability structure. Moreover, we construct three more infinite families of graphs which are closely related to convex polytopes, having an unbounded metric dimension.