Startseite Fault-tolerant resolvability of some graphs of convex polytopes
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Fault-tolerant resolvability of some graphs of convex polytopes

  • Sunny K. Sharma , Hassan Raza EMAIL logo und Vijay K. Bhat
Veröffentlicht/Copyright: 18. Juni 2023
Veröffentlichen auch Sie bei De Gruyter Brill

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 𝕎 of vertices of a connected graph G in which the vector of distances to the vertices in 𝕎 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 𝕎 ∖ ρ is also a resolving set for each ρ in 𝕎. In that case, 𝕎 is said to be a fault-tolerant resolving set. The fault-tolerant metric dimension of G is the minimum cardinality of such a set 𝕎. 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, №4, 108–122 (in Russian).


References

[1] Bǎca M., “Labellings of two classes of convex polytopes”, Util. Math., 34 (1988), 24–31.Suche in Google Scholar

[2] Basak M., Saha L., Das G. K., Tiwary K., “Fault-tolerant metric dimension of circulant graphs cn(1, 2, 3)”, Theor. Comput. Sci., 817 (2020), 66–79.Suche in Google Scholar

[3] Bashir H., Zahid Z., Kashif A., Zafar S., Liu J. B., “On 2-metric resolvability in rotationally-symmetric graphs”, J. Intell. Fuzzy Syst., 2021, 1–9.Suche in Google Scholar

[4] Beerloiva Z., Eberhard F., Erlebach T., Hall A., Hoffmann M., Mihalak M., Ram L., “Network discovery and verification”, IEEE J. Sel. Area Commun., 24 (2006), 2168–2181.Suche in Google Scholar

[5] Chartrand G., Eroh L., Johnson M. A., Oellermann O. R., “Resolvability in graphs and the metric dimension of a graph”, Discrete Appl. Math., 105 (2000), 99–113.Suche in Google Scholar

[6] Chartrand G., Saenpholphat V., Zhang R., “The independent resolving number of a graph”, Math. Bohem., 128 (2003), 379–393.Suche in Google Scholar

[7] Chartrand G., Zhang R., “The theory and applications of resolvability in graphs: a survey”, Congr. Numer., 160 (2003), 47–68.Suche in Google Scholar

[8] Chvatal V., “Mastermind”, Combinatorica, 3 (1983), 325–329.Suche in Google Scholar

[9] Garey M. R., Johnson D. S., Computers and Intractability: A Guide to the Theory of NP–Completeness, W. H. Freeman and Company, 1979.Suche in Google Scholar

[10] Guo X., Faheem M., Zahid Z., Nazeer W., Li J., “Fault-tolerant resolvability in some classes of line graphs”, Math. Probl. in Engineering, 4 (2020), 1–8.Suche in Google Scholar

[11] Harary F., Melter R. A., “On the metric dimension of a graph”, Ars Comb., 2 (1976), 191–195.Suche in Google Scholar

[12] Hernando C., Mora M., Slater R. J., Wood D. R., “Fault-tolerant metric dimension of graphs”, Ramanujan Math. Soc. Lect. Notes, Proc. Int. Conf. Convexity in Discrete Structures, 5, 2008, 81–85.Suche in Google Scholar

[13] Honkala I., Laihonen T., “On locating-dominating sets in infinite grids”, Eur. J. Comb., 27:2 (2006), 218–227.Suche in Google Scholar

[14] Javaid I., Salman M., Chaudhry M. A., Shokat S., “Fault-tolerance in resolvability”, Util. Math., 80 (2009), 263–275.Suche in Google Scholar

[15] Jesse G., “Metric dimension and pattern avoidance in graphs”, Discret. Appl. Math., 284 (2020), 1–7.Suche in Google Scholar

[16] Bensmail J., Inerney F. M., Nisse N., “Metric dimension: from graphs to oriented graphs”, Discret. Appl. Math., 323 (2020), 28–42.Suche in Google Scholar

[17] Khuller S., Raghavachari B., Rosenfeld A., “Landmarks in graphs”, Discrete Appl. Math., 70 (1996), 217–229.Suche in Google Scholar

[18] Rehman S. ur, Imran M., Javaid I., “On the metric dimension of arithmetic graph of a composite number”, Symmetry, 12:4, #607 (2020), 10 pp.Suche in Google Scholar

[19] Raza H., Hayat S., Imran M., Pan X. F., “Fault-tolerant resolvability and extremal structures of graphs”, Mathematics, 7:1, #78 (2019), 19 pp.Suche in Google Scholar

[20] Raza H., Hayat S., Pan X. F., “On the fault-tolerant metric dimension of convex polytopes”, Appl. Math. Comput., 339 (2018), 172–185.Suche in Google Scholar

[21] Raza H., Hayat S., Pan X. F., “On the fault-tolerant metric dimension of certain interconnection networks”, J. Appl. Math. Comput., 60:1 (2019), 517–535.Suche in Google Scholar

[22] Raza H., Liu J. B., Qu S., “On mixed metric dimension of rotationally symmetric graphs”, IEEE Access, 8 (2020), 11560–11569.Suche 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:1409.2695Suche in Google Scholar

[24] Sharma S. K., Bhat V. K., “Metric dimension of heptagonal circular ladder”, Discrete Math. Algorithms Appl., 13:1, #2050095 (2021), 17 pp.Suche in Google Scholar

[25] Sharma S. K., Bhat V. K., “Fault-tolerant metric dimension of two-fold heptagonal-nonagonal circular ladder”, DiscreteMath. Algorithms Appl., 14:3, #2150132 (2022).Suche in Google Scholar

[26] Sharma S. K., Bhat V. K., “On metric dimension of plane graphs 𝔉n, 𝔎n, and 𝔏n”, J. Algebra Comb. Discrete Struct. Appl., 8:3 (2021), 197–212.Suche in Google Scholar

[27] Sharma S. K., Bhat V. K., “Edge metric dimension and edge basis of one-heptagonal carbon nanocone networks”, IEEE Access, 10 (2022), 29558–29566.Suche in Google Scholar

[28] Siddiqui H. M. A., Hayat S., Khan A., Imran M., Razzaq A., Liu J. -B., “Resolvability and fault-tolerant resolvability structures of convex polytopes”, Theor. Comput. Sci., 796 (2019), 114–128.Suche in Google Scholar

[29] Slater P. J., “Leaves of trees”, Congr. Numer., 14 (1975), 549–559.Suche in Google Scholar

[30] Soderberg S., Shapiro H. S., “A combinatory detection problem”, Amer. Math. Mon., 70:10 (1963), 1066–1070.Suche in Google Scholar

[31] Stojmenovic I., “Direct interconnection networks”, Parallel and Distributed Computing Handbook, eds. Zomaya A. Y., McGraw-Hill, 1996, 537–567.Suche in Google Scholar

[32] Xuanlong M., She Y., “The metric dimension of the enhanced power graph of a finite group”, J. Algebra. Appl., 19:1, #2050020 (2020).Suche in Google Scholar

[33] Yuezhong Z., Hou L., Hou B., WuW., Du D., Gao S., “On the metric dimension of the folded n-cube”, Optim. Lett., 14:1 (2020), 249–257.Suche in Google Scholar

Received: 2022-03-14
Published Online: 2023-06-18
Published in Print: 2023-06-27

© 2023 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 30.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/dma-2023-0016/pdf?lang=de
Button zum nach oben scrollen