Locating number of biswapped networks. (English) Zbl 1517.68034
Summary: A biswapped network is an interconnection network in multiprocessor systems. These are networks of devices or processors and connections of these devices or processors. Here network can be expressed in the form of a graph. Devices or processors represent vertices, and the connection of devices or processors represents edges. The subset of the nodes set is called the resolving set if all the representations or codes are different w.r.t. this subset gives the information about the complete network. The minimum cardinality of the resolving set is called the locating number. In this article, we find the locating number of biswapped interconnection networks. In this paper, We compute the exact values of locating numbers for biswapped networks generated by different families of underlying basis networks like path, cycle, power of path, and complete. We have also given the bounds of locating number of any biswapped network generated by any underlying basis network that consists of its clusters.
MSC:
| 68M10 | Network design and communication in computer systems |
| 68R10 | Graph theory (including graph drawing) in computer science |
References:
| [1] | Abrishami, G., Henning, M. A., and Tavakoli, M., Local metric dimension for graphs with small clique numbers, Discr. Math.345(4) (2022) 112763. · Zbl 1482.05076 |
| [2] | Bensmail, J., Inerney, F. Mc and Nisse, N., Metric dimension: From graphs to oriented graphs, Discr. Appl. Math.346 (2019) 111-123. · Zbl 07515173 |
| [3] | Chartrand, G., Eroh, L., Johnson, M. A. and Oellermann, O. R., Resolvability in graphs and the metric dimension of a graph, Discr. Appl. Math.105(1-3) (2000) 99-113. · Zbl 0958.05042 |
| [4] | M. R. Garey and D. S. Johnson, Computers and intractability: A guide to the theory of NP-completeness, W. H. Freeman and Company, 1979. · Zbl 0411.68039 |
| [5] | Gupta, A. and Sarkar, B. K., The recursive and symmetrical optoelectronic network architecture: Biswapped network hyper hexa-cell, Arabian Journal for Science and Engineering43(12) (2018) 7635-7653. |
| [6] | Harary, F. and Melter, R. A., On the metric dimension of a graph, Ars Combinatoria2 (1976) 191-195. · Zbl 0349.05118 |
| [7] | Jiang, Z. and Polyanskii, N., On the metric dimension of Cartesian powers of a graph, Journal of Combinatorial Theory, Series A165 (2019) 1-14. · Zbl 1414.05251 |
| [8] | Khuller, S., Raghavachari, B. and Rosenfeld, A., Landmarks in graphs, Discr. Appl. Math.70(3) (1996) 217-229. · Zbl 0865.68090 |
| [9] | Maji, D., Ghorai, G. and Shami, F. A., Some new upper bounds for the — Index of graphs, J. Math.2022 (2022) Article ID 4346234, 13 pages. |
| [10] | Maji, D. and Ghorai, G., Computing F-index, coindex and Zagreb polynomials of the kth generalized transformation graphs, Heliyon6(12) (2020) e05781. |
| [11] | Maji, D. and Ghorai, G., A novel graph invariant: The third leap Zagreb index under several graph operations, Discr. Math., Algorithms and Applications11(5) (2019) 1950054. · Zbl 1426.05013 |
| [12] | Manuel, P., Bharati, R., Rajasingh, I. and Monica M, C., On minimum metric dimension of honeycomb networks, J. Discr. Algorithms6(1) (2008) 20-27. · Zbl 1159.05308 |
| [13] | Perc, M. and Szolnoki, A., Coevolutionary games-A mini review, Biosystems99(2) (2010) 109-125. |
| [14] | Raj, F. S. and George, A., On the metric dimension of HDN 3 and PHDN 3, in Proc. IEEE Int. Conf. Power, Control, Signals Instrum. Eng. (ICPCSI), Sep. 2017, pp. 1333-1336. |
| [15] | Sebő, A. and Tannier, E., On metric generators of graphs, Mathematics of Operations Research29(2) (2004) 383-393. · Zbl 1082.05032 |
| [16] | Shao, Z., Sheikholeslami, S. M., Wu, P. and Liu, AJ.-B., The metric dimension of some generalized Petersen graphs, Discrete Dynamics in Nature and Society2018 (2018) 1-10. · Zbl 1417.05054 |
| [17] | Shao, Z., Wu, P., Zhu, E. and Chen, L., On metric dimension in some hex derived networks, Sensors19(1) (2018) 94. |
| [18] | Slater, P. J., Leaves of trees, Proceeding of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium14 (1975) 549-559. · Zbl 0316.05102 |
| [19] | Soderberg, S. and Shapiro, H. S., A combinatory detection problem, The American Mathematical Monthly70(10) (1963) 1066. · Zbl 0123.00205 |
| [20] | Tsai, C.-H. and Chen, J.-C., Fault isolation and identification in general biswapped networks under the PMC diagnostic model, Theoretical Computer Science501 (2013) 62-71. · Zbl 1296.68020 |
| [21] | W. Wei and W. Xiao, Fault tolerance in the biswapped network, Lecture Notes in Computer Science, pp. 79-82. |
| [22] | W. Xiao, W. Chen, M. He, W. Wei and B. Parhami, Biswapped networks and their topological properties, Eighth ACIS International Conference on Software Engineering, Artificial Intelligence, Networking, and Parallel/Distributed Computing (SNPD 2007), Jul. 2007. |
| [23] | Xiao, W., Parhami, B., Chen, W., He, M. and Wei, W., Biswapped networks: A family of interconnection architectures with advantages over swapped or OTIS networks, International Journal of Computer Mathematics88(13) (2011) 2669-2684. · Zbl 1237.68025 |
| [24] | Yero, I. G., Kuziak, D. and Rodríguez-Velázquez, J. A., On the metric dimension of corona product graphs, Comput. Math. Appl.61(9) (2011) 2793-2798. · Zbl 1221.05252 |
| [25] | Yu, Y. and Wei, W., Load balancing on the biswapped network, 2009 Second International Conference on Intelligent Networks and Intelligent Systems, Nov. 2009. |
| [26] | Zhang, Y., Hou, L., Hou, B., Wu, W., Du, D.-Z. and Gao, S., On the metric dimension of the folded n-cube, Optimization Letters14(1) (2019) 249-257. · Zbl 1433.05106 |
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