Computing the metric dimension of wheel related graphs

Hafiz Muhammad Afzal Siddiqui, Muhammad Imran

Research output: Contribution to journalArticlepeer-review

30 Citations (Scopus)

Abstract

An ordered set W={w1,...,wk}V(G) of vertices of G is called a resolving set or locating set for G if every vertex is uniquely determined by its vector of distances 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 Wn, m,Qn and AWJ2n, respectively. We prove that these infinite classes of wheel related graphs have unbounded metric dimension. The study of an infinite class of convex polytopes generated by wheel, denoted by Qn also gives a negative answer to an open problem proposed by Imran et al. (2012) in [8]: 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.

Original languageEnglish
Pages (from-to)624-632
Number of pages9
JournalApplied Mathematics and Computation
Volume242
DOIs
Publication statusPublished - Sep 1 2014
Externally publishedYes

Keywords

  • Basis
  • Metric dimension
  • Resolving set
  • Wheel related graph

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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