An infinite class of convex polytopes with constant metric dimension

Muhammad Imran, A. Q. Baig

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

A family G of connected graphs is a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G.- The metric dimension of some classes of plane graphs has been determined in [3], [4], [5], [12], [14] and [18] while metric dimension of some families of convex polytopes has been studied in [8], [9], [10] and [11]and the following open problem was raised in [11]. Open Problem [11]: Let G be the graph of a convex polytope which is obtained by joining the graph of two different convex polytopes G 1 and G 2 (such that the outer cycle of G 1 is the inner cycle of G 2) both having constant metric dimension. Is it the case that G will always have the constant metric dimension? In this paper, we extend this study to an infinite class of convex polytopes which is obtained as a combination of graph of an antiprism A n [1] and graph of convex polytope Q n [2] such that the outer cycle of A n is the inner cycle of Q n. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension. Note that the problem of determining whether (Hm(G) < k is an JVP-complete problem [7].

Original languageEnglish
Pages (from-to)3-9
Number of pages7
JournalJournal of Combinatorial Mathematics and Combinatorial Computing
Volume81
Publication statusPublished - May 2012
Externally publishedYes

Keywords

  • Basis
  • Convex polytope
  • Metric dimension
  • Plane graph
  • Resolving set

ASJC Scopus subject areas

  • General Mathematics

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