AIMS Mathematics, 2020, 5(6): 6221-6232. doi: 10.3934/math.2020400

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On the metric basis in wheels with consecutive missing spokes

1 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan 60800, Pakistan
2 Informetrics Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3 Faculty of Mathematics & Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

If $G$ is a connected graph, the $distance$ $d(u, v)$ between two vertices $u, v \in V(G)$ is the length of a shortest path between them. Let $W = \{w_1,w_2, \dots ,w_k\}$ be an ordered set of vertices of $G$ and let $v$ be a vertex of $G$. The $representation$ $r(v|W)$ of $v$ with respect to $W$ is the k-tuple $(d(v,w_1), d(v,w_2), \dots , d(v,w_k))$. $W$ 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 $\beta(G)$. The metric dimension of some wheel related graphs is studied recently by Siddiqui and Imran. In this paper, we study the metric dimension of wheels with $k$ consecutive missing spokes denoted by $W(n,k)$. We compute the exact value of the metric dimension of $W(n,k)$ which shows that wheels with consecutive missing spokes have unbounded metric dimensions. It is natural to ask for the characterization of graphs with an unbounded metric dimension. The exchange property for resolving a set of $W(n,k)$ has also been studied in this paper and it is shown that exchange property of the bases in a vector space does not hold for minimal resolving sets of wheels with $k$-consecutive missing spokes denoted by $W(n,k)$.
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References

1. G. Abbas, U. Ali, M. Munir, et al. Power graphs and exchange property for resolving sets, Open Math., 17 (2019), 1303-1309.    

2. D. L. Boutin, Determining sets, resolving sets and the exchange property, Graphs Combin., 25 (2009), 789-806.    

3. P. S. Buczkowski, G. Chartrand, C. Poisson, et al. On k-dimensional graphs and their bases, Periodica Math. Hung., 46 (2003), 9-15.    

4. J. Caceres, C. Hernando, M. Mora, et al. On the metric dimension of cartesian product of graphs, SIAM J. Disc. Math., 2 (2007), 423-441.

5. P. J. Cameron, J. H. Van Lint, Designs, Graphs, Codes and their Links, London Mathematical Society Student Texts 22, Cambridge University Press, Cambridge, 1991.

6. G. Chartrand, L. Eroh, M. A. Johnson, et al. Resolvability in graphs and metric dimension of a graph, Disc. Appl. Math., 105 (2000), 99-113.    

7. F. M. Dong, Y. P. Liu, All wheels with two missing consecutive spokes are chromatically unique, Disc. Math., 184 (1998), 71-85.    

8. M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP- Completeness, Freeman, New York, 1979.

9. F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Combin., 2 (1976), 191-195.

10. M. Imran, A. Q. Baig, M. K. Shafiq, et al. On metric dimension of generlized Petersen graphs P(n, 3), Ars Combin., 117 (2014), 113-130.

11. M. Imran, A. Q. Baig, S. A. Bokhary, et al. On the metric dimension of circulant graphs, Appl. Math. Lett., 25 (2012), 320-325.    

12. M. Imran, S. A. Bokhary, A. Q. Baig, On families of convex polytopes with constant metric dimension, Comput. Math. Appl., 60 (2010), 2629-2638.    

13. M. Imran, A. Ahmad, S. A. Bokhary, et al. On classes of regular graphs with constant metric dimension, Acta Math. Scientia, 33 (2013), 187-206.    

14. M. Imran, H. M. A. Siddiqui, Computing the metric dimension of convex polytopes generated by wheel related graphs, Acta Math. Hungar., 149 (2016), 10-30.    

15. I. Javaid, M. T. Rahim, K. Ali, Families of regular graphs with constant metric dimension, Utilitas Math., 75 (2008), 21-33.

16. S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Disc. Appl. Math., 70 (1996), 217-229.    

17. S. Khuller, B. Raghavachari, A. Rosenfeld, Localization in graphs, Technical Report CS-TR-3326, University of Maryland at College Park, 1994.

18. R. A. Melter, I. Tomescu, Metric bases in digital geometry, Comput. Vision Graphics, Image Processing, 25 (1984), 113-121.    

19. O. R. Oellermann, J. Peters-Fransen, Metric dimension of Cartesian products of graphs, Utilitas Math., 69 (2006), 33-41.

20. M. Salman, I. Javaid, M. A. Chaudhry, Resolvability in circulant graphs, Acta Math. Sin. Engl. Ser., 28 (2012), 1851-1864.    

21. A. Sebö, E. Tannier, On metric generators of graphs, Math. Oper. Res., 29(2004),383-393.    

22. H. M. A. Siddiqui, M. Imran, Computing the metric dimension of wheel related graphs, Appl. Math. Comput., 242 (2014), 624-632.

23. P. J. Slater, Leaves of trees, Congress. Numer., 14 (1975), 549-559.

24. P. J. Slater, Dominating and reference sets in graphs, J. Math. Phys. Sci., 22 (1988), 445-455.

25. I. Tomescu, M. Imran, On metric and partition dimensions of some infinite regular graphs, Bull. Math. Soc. Sci. Math. Roumanie, 52 (2009), 461-472.

26. I. Tomescu, I. Javaid, On the metric dimension of the Jahangir graph, Bull. Math. Soc. Sci. Math. Roumanie, 50 (2007), 371-376.

27. U. Ali, S. A. Bokhary, K. Wahid, et al. On resolvability of a graph associated to a finite vector space, Journal of Algebra and Its Applications, 18 (2018), 1950028.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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