Abstract
In the world of virtual and connected networks, the study of network’s computing capabilities through graph embedding has grown in prominence. In order to implement algorithms created for the guest graph in the host graph, embedding involves simulating one architecture, called the guest, into another, called the host. In this paper, we have obtained the optimal wirelength of embedding \((K_9-C_9)^n\) into Mesh \(M(3^n,3^n)\), generalized book graph \({\text {GB}}[9,3^{n-1},3^{n-1}]\) and extended firecracker tree \({\text {EF}}_{9^n}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Parthiban N, Ryan J, Rajasingh I, Rajan RS, Rani LN (2017) Exact wirelength of embedding chord graph into tree-based architectures. Int J Netw Virtual Organ 17(1):76–87
Bhagat S, Cormode G, Muthukrishnan S (2011) Node classification in social networks. Social network data analytics. Springer, pp 115–148
Hauswirth M, Schmidt R (2005) An overlay network for resource discovery in grids. In: IEEE International Workshop on Database and Expert Systems Applications (DEXA’05), pp 343–348
Jungnickel D, Jungnickel D (2005) Graphs, networks and algorithms, vol 3. Springer
Dong Q, Yang X, Wang D (2010) Embedding paths and cycles in 3-ary n-cubes with faulty nodes and links. Inf Sci 180(1):198–208
Arockiaraj M, Shalini AJ, Delaila JN (2022) Embedding algorithm of spined cube into grid structure and its wirelength computation. Theor Comput Sci 905:69–86
Miller M, Rajan RS, Parthiban N, Rajasingh I (2015) Minimum linear arrangement of incomplete hypercubes. Comput J 58(2):331–337
Abraham J, Arockiaraj M (2021) Minimum layout of hypercubes and folded hypercubes into the prism over caterpillars. Int J Comput Aided Eng Technol 14(4):520–529
Fan W-B, Fan J-X, Lin C-K, Wang Y, Han Y-J, Wang R-C (2019) Optimally embedding 3-ary n-cubes into grids. J Comput Sci Technol 34(2):372–387
Bae MM, Bose B (2003) Edge disjoint Hamiltonian cycles in k-ary n-cubes and hypercubes. IEEE Trans Comput 52(10):1271–1284
Yang M (2009) Path embedding in star graphs. Appl Math Comput 207(2):283–291
Xu J-M, Ma M (2006) Cycles in folded hypercubes. Appl Math Lett 19(2):140–145
Fan W, Wang Y, Sun J, Han Z, Li P, Wang R (2019) Fault-tolerant cycle embedding into 3-ary n-cubes with structure faults. In: 2019 IEEE International Conference on Parallel & Distributed Processing with Applications, Big Data & Cloud Computing, Sustainable Computing & Communications, Social Computing & Networking, pp 451–457
Rajan RS, Parthiban N, Rajalaxmi T (2015) Embedding of recursive circulants into certain necklace graphs. Math Comput Sci 9(2):253–263
Rajasingh I, Manuel P, Arockiaraj M, Rajan B (2013) Embeddings of circulant networks. J Comb Optim 26(1):135–151
Arockiaraj M, Abraham J, Shalini AJ (2019) Node set optimization problem for complete Josephus cubes. J Comb Optim 38(4):1180–1195
Tang Z (2022) Optimal embedding of hypercube into cylinder. Theor Comput Sci 923:327–336
Bezrukov SL, Das SK, Elsässer R (2000) An edge-isoperimetric problem for powers of the Petersen graph. Ann Comb 4(2):153–169
Bezrukov SL, Chavez JD, Harper LH, Röttger M, Schroeder U (1998) Embedding of hypercubes into grids. In: Mathematical foundations of computer science. Lecture notes in computer science, vol 1450, pp 693–701
Manuel PD, Rajasingh I, Rajan B, Mercy H (2009) Exact wirelength of hypercubes on a grid. Discret Appl Math 157(7):1486–1495
Arockiaraj M, Manuel PD, Rajasingh I, Rajan B (2011) Wirelength of 1-fault Hamiltonian graphs into wheels and fans. Inf Process Lett 111(18):921–925
Afiya S, Rajesh M (2022) Embedding k 9-c 9 n into certain necklace graphs. In: International Conference on Big Data and Cloud Computing, pp 359–369. Springer
Bezrukov SL, Bulatovic P, Kuzmanovski N (2018) New infinite family of regular edge-isoperimetric graphs. Theor Comput Sci 721:42–53
Ahlswede R, Cai N (1997) General edge-isoperimetric inequalities, part II: a local-global principle for lexicographical solutions. Eur J Comb 18(5):479–489
Afiya S, Rajesh M (2023) Minla of (k 9-c 9) n and its optimal layout into certain trees. J Supercomput 79:1–13
Rajan RS, Rajasingh I, Arockiaraj M, Rajalaxmi T, Mahavir B (2017) Embedding of hypercubes into generalized books. Ars Comb 135:133–151
Chen W-C, Lu H-I, Yeh Y-N (1997) Operations of interlaced trees and graceful trees. Southeast Asian Bull Math 21(4):337–348
Acknowledgements
We would like to thank Dr. Indra Rajasingh, Adjunct Professor, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, for her valuable suggestions.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
Syeda Afiya and Rajesh M conceived of the presented idea and developed the theory. All authors discussed the results and contributed to the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
We declare no conflict of interest.
Ethical approval
We adhered to the accepted ethical standards of a genuine research study.
Consent to participate
The author did not undertake work that involved Human or Animal participants.
Consent for publication
Yes.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Afiya, S., Rajesh, M. Embedding \((K_9-C_9)^n\) into interconnection networks. J Supercomput 80, 13475–13491 (2024). https://doi.org/10.1007/s11227-024-05941-0
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11227-024-05941-0