Resolvability and fault-tolerant resolvability structures of convex polytopes

Hafiz Muhammad Afzal Siddiqui, Sakander Hayat, Asad Khan, Muhammad Imran, Ayesha Razzaq, Jia Bao Liu

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)114-128
Number of pages15
JournalTheoretical Computer Science
Volume796
DOIs
Publication statusPublished - Dec 3 2019

Keywords

  • Convex polytopes
  • Fault-tolerant metric dimension
  • Geometric graphs
  • Metric dimension
  • NP-complete problems

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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