Abstract
The fault-tolerant resolvability is an extension of metric resolvability in graphs with several intelligent systems applications, for example, network optimization, robot navigation, and sensor networking. The graphs of convex polytopes, which are rotationally symmetric, are essential in intelligent networks due to the uniform rate of data transformation to all nodes. A resolving set is an ordered set đ of vertices of a connected graph G in which the vector of distances to the vertices in đ uniquely determines all the vertices of the graph G. The minimum cardinality of a resolving set of G is known as the metric dimension of G. If đ â Ï is also a resolving set for each Ï in đ. In that case, đ is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of G is the minimum cardinality of such a set đ. The metric dimension and the fault-tolerant metric dimension for three families of convex polytope graphs are studied. Our main results affirm that three families, as mentioned above, have constant fault-tolerant resolvability structures.
Originally published in Diskretnaya Matematika (2023) 34, â4, 108â122 (in Russian).
References
[1] BÇca M., âLabellings of two classes of convex polytopesâ, Util. Math., 34 (1988), 24â31.Search in Google Scholar
[2] Basak M., Saha L., Das G. K., Tiwary K., âFault-tolerant metric dimension of circulant graphs cn(1, 2, 3)â, Theor. Comput. Sci., 817 (2020), 66â79.Search in Google Scholar
[3] Bashir H., Zahid Z., Kashif A., Zafar S., Liu J. B., âOn 2-metric resolvability in rotationally-symmetric graphsâ, J. Intell. Fuzzy Syst., 2021, 1â9.Search in Google Scholar
[4] Beerloiva Z., Eberhard F., Erlebach T., Hall A., Hoffmann M., Mihalak M., Ram L., âNetwork discovery and verificationâ, IEEE J. Sel. Area Commun., 24 (2006), 2168â2181.Search in Google Scholar
[5] Chartrand G., Eroh L., Johnson M. A., Oellermann O. R., âResolvability in graphs and the metric dimension of a graphâ, Discrete Appl. Math., 105 (2000), 99â113.Search in Google Scholar
[6] Chartrand G., Saenpholphat V., Zhang R., âThe independent resolving number of a graphâ, Math. Bohem., 128 (2003), 379â393.Search in Google Scholar
[7] Chartrand G., Zhang R., âThe theory and applications of resolvability in graphs: a surveyâ, Congr. Numer., 160 (2003), 47â68.Search in Google Scholar
[8] Chvatal V., âMastermindâ, Combinatorica, 3 (1983), 325â329.Search in Google Scholar
[9] Garey M. R., Johnson D. S., Computers and Intractability: A Guide to the Theory of NPâCompleteness, W. H. Freeman and Company, 1979.Search in Google Scholar
[10] Guo X., Faheem M., Zahid Z., Nazeer W., Li J., âFault-tolerant resolvability in some classes of line graphsâ, Math. Probl. in Engineering, 4 (2020), 1â8.Search in Google Scholar
[11] Harary F., Melter R. A., âOn the metric dimension of a graphâ, Ars Comb., 2 (1976), 191â195.Search in Google Scholar
[12] Hernando C., Mora M., Slater R. J., Wood D. R., âFault-tolerant metric dimension of graphsâ, Ramanujan Math. Soc. Lect. Notes, Proc. Int. Conf. Convexity in Discrete Structures, 5, 2008, 81â85.Search in Google Scholar
[13] Honkala I., Laihonen T., âOn locating-dominating sets in infinite gridsâ, Eur. J. Comb., 27:2 (2006), 218â227.Search in Google Scholar
[14] Javaid I., Salman M., Chaudhry M. A., Shokat S., âFault-tolerance in resolvabilityâ, Util. Math., 80 (2009), 263â275.Search in Google Scholar
[15] Jesse G., âMetric dimension and pattern avoidance in graphsâ, Discret. Appl. Math., 284 (2020), 1â7.Search in Google Scholar
[16] Bensmail J., Inerney F. M., Nisse N., âMetric dimension: from graphs to oriented graphsâ, Discret. Appl. Math., 323 (2020), 28â42.Search in Google Scholar
[17] Khuller S., Raghavachari B., Rosenfeld A., âLandmarks in graphsâ, Discrete Appl. Math., 70 (1996), 217â229.Search in Google Scholar
[18] Rehman S. ur, Imran M., Javaid I., âOn the metric dimension of arithmetic graph of a composite numberâ, Symmetry, 12:4, #607 (2020), 10 pp.Search in Google Scholar
[19] Raza H., Hayat S., Imran M., Pan X. F., âFault-tolerant resolvability and extremal structures of graphsâ, Mathematics, 7:1, #78 (2019), 19 pp.Search in Google Scholar
[20] Raza H., Hayat S., Pan X. F., âOn the fault-tolerant metric dimension of convex polytopesâ, Appl. Math. Comput., 339 (2018), 172â185.Search in Google Scholar
[21] Raza H., Hayat S., Pan X. F., âOn the fault-tolerant metric dimension of certain interconnection networksâ, J. Appl. Math. Comput., 60:1 (2019), 517â535.Search in Google Scholar
[22] Raza H., Liu J. B., Qu S., âOn mixed metric dimension of rotationally symmetric graphsâ, IEEE Access, 8 (2020), 11560â11569.Search in Google Scholar
[23] Salman M., Javaid I., Chaudhry M. A., Minimum fault-tolerant, local and strong metric dimension of graphs, 2014, 19 pp., arXiv: arXiv:1409.2695Search in Google Scholar
[24] Sharma S. K., Bhat V. K., âMetric dimension of heptagonal circular ladderâ, Discrete Math. Algorithms Appl., 13:1, #2050095 (2021), 17 pp.Search in Google Scholar
[25] Sharma S. K., Bhat V. K., âFault-tolerant metric dimension of two-fold heptagonal-nonagonal circular ladderâ, DiscreteMath. Algorithms Appl., 14:3, #2150132 (2022).Search in Google Scholar
[26] Sharma S. K., Bhat V. K., âOn metric dimension of plane graphs đn, đn, and đnâ, J. Algebra Comb. Discrete Struct. Appl., 8:3 (2021), 197â212.Search in Google Scholar
[27] Sharma S. K., Bhat V. K., âEdge metric dimension and edge basis of one-heptagonal carbon nanocone networksâ, IEEE Access, 10 (2022), 29558â29566.Search in Google Scholar
[28] Siddiqui H. M. A., Hayat S., Khan A., Imran M., Razzaq A., Liu J. -B., âResolvability and fault-tolerant resolvability structures of convex polytopesâ, Theor. Comput. Sci., 796 (2019), 114â128.Search in Google Scholar
[29] Slater P. J., âLeaves of treesâ, Congr. Numer., 14 (1975), 549â559.Search in Google Scholar
[30] Soderberg S., Shapiro H. S., âA combinatory detection problemâ, Amer. Math. Mon., 70:10 (1963), 1066â1070.Search in Google Scholar
[31] Stojmenovic I., âDirect interconnection networksâ, Parallel and Distributed Computing Handbook, eds. Zomaya A. Y., McGraw-Hill, 1996, 537â567.Search in Google Scholar
[32] Xuanlong M., She Y., âThe metric dimension of the enhanced power graph of a finite groupâ, J. Algebra. Appl., 19:1, #2050020 (2020).Search in Google Scholar
[33] Yuezhong Z., Hou L., Hou B., WuW., Du D., Gao S., âOn the metric dimension of the folded n-cubeâ, Optim. Lett., 14:1 (2020), 249â257.Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
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- Frontmatter
- On scatter properties of modular addition operation over imprimitivity systems of the translation group of the binary vector space
- On a number of particles in a marked set of cells in a general allocation scheme
- On the equality problem of finitely generated classes of exponentially-polynomial functions
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- 10.1515/dma-2023-0018