On families of convex polytopes with constant metric dimension

Imran, Muhammad; Bokhary, Syed Ahtsham Ul Haq; Baig, Abdul Qudair

doi:10.1016/j.camwa.2010.08.090

Cited by 61 publications

(87 citation statements)

References 8 publications

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“…If all graphs in F have the same metric dimension (which does not depend on n), F is called a family with constant metric dimension [12]. Some classes of regular graphs with constant metric dimension have been studied in [1,10] recently while metric dimension of some classes of convex polytopes has been determined in [7] and [9]. Other families of graphs have unbounded metric dimension: if W n denotes a wheel with n spokes and J 2n the graph deduced from the wheel W 2n by alternately deleting n spokes, thenˇ.W n / D b [18] for every n 4.…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On the metric dimension of barcycentric subdivision of Cayley graphs $Cay(Z_{n}\oplus Z_{m})$

Imran¹,

Ahmad²,

Al-Mushayt³

et al. 2015

Miskolc Math Notes

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Abstract. Let W D fw 1 ; w 2 ; : : : ; w k g be an ordered set of vertices of G and let v be a vertex of G. The representation r.vjW / of v with respect to W is the k-tuple .d.v; w 1 /; d.v; w 2 /; : : : ; d.v; w k //. W is called a resolving set or a locating set if every vertex of G is uniquely identified by its distances from the vertices of W , or equivalently, if distinct vertices of G have distinct representations with respect to W . A resolving set of minimum cardinality is called a metric basis for G and this cardinality is the metric dimension of G, denoted by d im.G/. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). In this paper, we study the metric dimension of barycentric subdivision of Cayley graphs C ay.Z n˚Zm /. We prove that these subdivisions of Cayley graphs have constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of Cayley graphs C ay.Z n˚Zm /.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

On the metric dimension of barcycentric subdivision of Cayley graphs $Cay(Z_{n}\oplus Z_{m})$

Imran¹,

Ahmad²,

Al-Mushayt³

et al. 2015

Miskolc Math Notes

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“…Also generalized Petersen graphs P(n, 2), antiprisms A n and circulant graphs C 2 n are families of graphs with constant metric dimension [16]. Some classes of regular graphs with constant metric dimension have been studied in [1,3,10] recently while metric dimension of some classes of convex polytopes has been determined in [11] and [13]. The metric dimension of graphs with pendent edges has been investigated in [15].…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

“…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) [5]. The concepts of resolving set and metric basis have previously appeared in the literature (see [1][2][3][4][5][6][7][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]). …”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Metric Dimension and Exchange Property for Resolving Sets in Rotationally-Symmetric Graphs

Naeem¹,

Imran²

2014

Appl. Math. Inf. Sci.

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Abstract:Metric dimension or location number is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs G n : F = (G n ) n≥1 depending on n as follows: the order |V (G)| = ϕ(n) and lim n→∞ ϕ(n) = ∞. If there exists a constant C > 0 such that dim(G n ) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension, otherwise F has unbounded metric dimension. If all graphs in F have the same metric dimension (which does not depend on n), F is called a family with constant metric dimension.In this paper, we study the metric dimension of quasi flower snarks, generalized antiprism and cartesian product of square cycle and path. We prove that these classes of graphs have constant or bounded metric dimension. It is natural to ask for characterization of graphs classes with respect to the nature of their metric dimension. It is also shown that the exchange property of the bases in a vector space does not hold for minimal resolving sets of quasi flower snarks, generalized prism and generalized antiprism.

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“…Applications of this invariant to the navigation of robots in networks are discussed in [8] and applications to chemistry in [2] while applications to problems of pattern recognition and image processing, some of which involve the use of hierarchical data structures are given in [9]. In [4,5,6 …”

Section: Introductionmentioning

confidence: 99%