Year 2016, Volume 3 , Issue 2, Pages 45 - 59 2016-05-15

Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let $\mathcal{F}$ be a family of connected graphs $G_{n}$ : $\mathcal{F} = (G_{n})_{n}\geq 1$ depending on $n$ as follows: the order $|V(G)| = \varphi(n)$ and $\lim\limits_{n\rightarrow \infty}\varphi(n)=\infty$. If there exists a constant $C > 0$ such that $dim(G_{n}) \leq C$ for every $n \geq 1$ then we shall say that $\mathcal{F}$ has bounded metric dimension, otherwise $\mathcal{F}$ has unbounded metric dimension. If all graphs in $\mathcal{F}$ have the same metric dimension, then $\mathcal{F}$ is called a family of graphs with constant metric dimension.

In this paper, we study the metric dimension of some classes of convex polytopes which are rotationally-symmetric. It is shown that these classes of convex polytoes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.

Metric dimension, Basis, Resolving set, Prism, Antiprism, Convex polytopes
  • [1] M. Baca, Labelings of two classes of convex polytopes, Utilitas Math. 34 (1988) 24–31.
  • [2] M. Baca, On magic labellings of convex polytopes, Ann. Disc. Math. 51 (1992) 13–16.
  • [3] P. S. Buczkowski, G. Chartrand, C. Poisson, P. Zhang, On k-dimensional graphs and their bases, Period. Math. Hungar. 46(1) (2003) 9–15.
  • [4] J. Caceres, C. Hernando, M. Mora, I. M. Pelayo, M. L. Puertas, C. Seara, D. R. Wood, On the metric dimension of cartesian products of graphs, SIAM J. Discrete Math. 21(2) (2007) 423–441.
  • [5] G. Chartrand, L. Eroh, M. A. Johnson, O. R. Oellermann, Resolvability in graphs and metric dimension of a graph, Discrete Appl. Math. 105(1-3) (2000) 99–113.
  • [6] M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NPCompleteness, New York: wh freeman, 1979.
  • [7] F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Combinatoria. 2 (1976) 191–195.
  • [8] C. Hernando, M. Mora, I. M. Pelayo, C. Seara, J. Caceres, M. L. Puertas, On the metric dimension of some families of graphs, Electron. Notes Discrete Math. 22 (2005) 129–133.
  • [9] M. Imran, A. Q. Baig, A. Ahmad, Families of plane graphs with constant metric dimension, Util. Math. 88 (2012) 43–57.
  • [10] M. Imran, A. Q. Baig, M. K. Shafiq, A. Semanicová–Fenovcíková, Classes of convex polytopes with constant metric dimension, Util. Math. 90 (2013) 85–99.
  • [11] I. Javaid, M. T. Rahim, K. Ali, Families of regular graphs with constant metric dimension, Util. Math. 75 (2008) 21–33.
  • [12] M. A. Johnson, Structure-activity maps for visualizing the graph variables arising in drug design, J. Biopharm. Stat. 3(2) (1993) 203–236.
  • [13] E. Jucovic, Convex polyhedra, Bratislava:Veda, 1981.
  • [14] S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math. 70(3) (1996) 217–229.
  • [15] S. Khuller, B. Raghavachari, A. Rosenfeld, Localization in graphs, 1998.
  • [16] R. A. Melter, I. Tomescu, Metric bases in digital geometry, Comput. Vision Graphics Image Process. 25(1) (1984) 113–121.
  • [17] J. Peters-Fransen, R. O. Oellermann, The metric dimension of Cartesian products of graphs, Util. Math. 69 (2006) 33–41.
  • [18] P. J. Slater, Leaves of trees, Congr. Numer. 14 (1975) 549–559.
  • [19] P. J. Slater, Dominating and reference sets in a graph, J. Math. Phys. Sci. 22(4) (1988) 445–455.
  • [20] I. Tomescu, M. Imran, On metric and partition dimensions of some infinite regular graphs, Bull. Math. Soc. Sci. Math. Roumanie. 52(100)(4) (2009) 461–472.
  • [21] I. Tomescu, I. Javaid, On the metric dimension of the Jahangir graph, Bull. Math. Soc. Sci. Math. Roumanie. 50(98)(4) (2007) 371–376.
Subjects Engineering
Journal Section Articles
Authors

Author: Muhammad Imran

Author: Syed Ahtsham Ul Haq Bokhary

Author: A. Q. Baig

Dates

Publication Date : May 15, 2016

Bibtex @research article { jacodesmath285342, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2016}, volume = {3}, pages = {45 - 59}, doi = {10.13069/jacodesmath.47485}, title = {On the metric dimension of rotationally-symmetric convex polytopes}, key = {cite}, author = {Imran, Muhammad and Bokhary, Syed Ahtsham Ul Haq and Baig, A. Q.} }
APA Imran, M , Bokhary, S , Baig, A . (2016). On the metric dimension of rotationally-symmetric convex polytopes . Journal of Algebra Combinatorics Discrete Structures and Applications , 3 (2) , 45-59 . DOI: 10.13069/jacodesmath.47485
MLA Imran, M , Bokhary, S , Baig, A . "On the metric dimension of rotationally-symmetric convex polytopes" . Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2016 ): 45-59 <https://dergipark.org.tr/en/pub/jacodesmath/issue/27121/285342>
Chicago Imran, M , Bokhary, S , Baig, A . "On the metric dimension of rotationally-symmetric convex polytopes". Journal of Algebra Combinatorics Discrete Structures and Applications 3 (2016 ): 45-59
RIS TY - JOUR T1 - On the metric dimension of rotationally-symmetric convex polytopes AU - Muhammad Imran , Syed Ahtsham Ul Haq Bokhary , A. Q. Baig Y1 - 2016 PY - 2016 N1 - doi: 10.13069/jacodesmath.47485 DO - 10.13069/jacodesmath.47485 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 45 EP - 59 VL - 3 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.47485 UR - https://doi.org/10.13069/jacodesmath.47485 Y2 - 2020 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications On the metric dimension of rotationally-symmetric convex polytopes %A Muhammad Imran , Syed Ahtsham Ul Haq Bokhary , A. Q. Baig %T On the metric dimension of rotationally-symmetric convex polytopes %D 2016 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 3 %N 2 %R doi: 10.13069/jacodesmath.47485 %U 10.13069/jacodesmath.47485
ISNAD Imran, Muhammad , Bokhary, Syed Ahtsham Ul Haq , Baig, A. Q. . "On the metric dimension of rotationally-symmetric convex polytopes". Journal of Algebra Combinatorics Discrete Structures and Applications 3 / 2 (May 2016): 45-59 . https://doi.org/10.13069/jacodesmath.47485
AMA Imran M , Bokhary S , Baig A . On the metric dimension of rotationally-symmetric convex polytopes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016; 3(2): 45-59.
Vancouver Imran M , Bokhary S , Baig A . On the metric dimension of rotationally-symmetric convex polytopes. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016; 3(2): 45-59.