Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-17T08:21:20.445Z Has data issue: false hasContentIssue false
  • English
  • Français

ISOLATED SCATTERING NUMBER OF SPLIT GRAPHS AND GRAPH PRODUCTS

Part of: Operations research and management science Graph theory Circuits, networks

Published online by Cambridge University Press:  12 April 2017

FENGWEI LI ,
QINGFANG YE  and
XIAOYAN ZHANG
FENGWEI LI*
Affiliation:
Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, PR China email [email protected], [email protected]
QINGFANG YE
Affiliation:
Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, PR China email [email protected], [email protected]
XIAOYAN ZHANG
Affiliation:
School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210023, PR China email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Computer or communication networks are so designed that they do not easily get disrupted under external attack. Moreover, they are easily reconstructed when they do get disrupted. These desirable properties of networks can be measured by various parameters, such as connectivity, toughness and scattering number. Among these parameters, the isolated scattering number is a comparatively better parameter to measure the vulnerability of networks. In this paper we first prove that for split graphs, this number can be computed in polynomial time. Then we determine the isolated scattering number of the Cartesian product and the Kronecker product of special graphs and special permutation graphs.

Keywords

isolated scattering numbersplit graphsubmodular functionKronecker productCartesian product

MSC classification

Primary: 90B18: Communication networks
Secondary: 94C15: Applications of graph theory 05C40: Connectivity 05C90: Applications
Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 3-4 , April 2017 , pp. 350 - 358
Copyright
© 2017 Australian Mathematical Society 

References

Alon, N. and Lubetzky, E., “Independent set in tensor graph powers”, J. Graph Theory 54 (2007) 73–87; doi:10.1002/jgt.20194.Google Scholar
Arika, T. and Shibata, Y., “Optimal design of diagnosable systems on networks constructed by graph operations”, Electron. Commun. Japan 85 (2002) 1–9; doi:10.1002/ecjc.1089.Google Scholar
Barefoot, C. A., Entringer, R. and Swart, H., “Vulnerability in graphs – A comparative survey”, J. Combin. Math. Combin. Comput. 1 (1987) 12–22; https://www.researchgate.net/publication/266002676.Google Scholar
Bondy, J. A. and Murty, U. S. R., Graph theory with applications (North-Holland, New York, 1976); http://101.96.10.59/www.iro.umontreal.ca/∼hahn/IFT3545/GTWA.pdf.Google Scholar
Bres̆ar, B., Imrich, W., Klavz̆ar, S. and Zmazek, B., “Hypercubes as direct products”, SIAM J. Discrete Math. 18 (2005) 778–786; doi:10.1137/S0895480103438358.Google Scholar
Chartrand, G. and Harary, F., “Planar permutation graphs”, Ann. Inst. H. Poincaré Sec. B (N.S.) 3 (1967) 433–438; https://eudml.org/doc/76875.Google Scholar
Chvátal, V., “Tough graphs and Hamiltonian circuits”, Discrete Math. 5 (1973) 215–228; doi:10.1016/j.disc.2006.03.011.CrossRefGoogle Scholar
Cozzens, M., Moazzami, D. and Stueckle, S., “The tenacity of a graph”, in: Proceedings of 7th International Conference on the Theory and Applications of Graphs (Wiley, New York, 1995) 1111–1122; http://101.96.10.59/www.iro.umontreal.ca/∼hahn/IFT3545/GTWA.pdf.Google Scholar
Foldes, S. and Hammer, P. L., “Split graphs having Dilworth number two”, Can. J. Math. 29 (1977) 666–672; doi:10.4153/CJM-1977-069-1.Google Scholar
Ghozati, S. A., “A finite automata approach to modeling the cross product of interconnection networks”, Math. Comput. Modelling 30 (1999) 185–200; doi:10.1016/S0895-7177(99)00173-9.Google Scholar
Grötschel, M., Lovász, L. and Schrijver, A., Geometric algorithms and combinatorial optimization (Springer, Berlin, 1988); http://www.apress.com/cn/book/9783642978814#otherversion=9783642978838.CrossRefGoogle Scholar
Jung, H. A., “On maximal circuits in finite graphs”, Ann. Discrete Math. 3 (1978) 129–144; doi:10.1016/S0167-5060(08)70503-X.CrossRefGoogle Scholar
Lammprey, R. H. and Barnes, B. H., “Products of graphs and applications”, Model. Simul. 5 (1974) 1119–1123; http://www.ams.org/mathscinet-getitem?mr=449900.Google Scholar
Li, F. W., “Isolated rupture degree of trees and gear graphs”, Neural Network World 25 (2015) 287–300; doi:10.14311/NNW.2015.25.015.Google Scholar
Li, F. W., “On isolated rupture degree of graphs”, Util. Math. 96 (2015) 33–47;https://www.researchgate.net/publication/292526797.Google Scholar
Li, Y. K., Zhang, S. G. and Li, X. L., “Rupture degree of graphs”, Int. J. Comput. Math. 82 (2005) 793–803; doi:10.1080/00207160412331336062.CrossRefGoogle Scholar
Mamut, A. and Vumar, E., “Vertex vulnerability parameters of Kronecker products of complete graphs”, Inform. Process. Lett. 106 (2008) 258–262; doi:10.1016/j.ipl.2007.12.002.Google Scholar
Quinn, M. J., Parallel computing: theory and practice (McGraw-Hill, New York, 1994);http://dl.acm.org/citation.cfm?id=174770.Google Scholar
Schrijver, A., “A combinatorial algorithm minimizing submodular functions in strongly polynomial time”, J. Combin. Theory Ser. B 80 (2000) 346–355; doi:10.1006/jctb.2000.1989.CrossRefGoogle Scholar
Wang, S. Y., Yang, Y. X., Lin, S. W., Li, J. and Hu, Z. M., “The isolated scattering number of graphs”, Acta Math. Sinica 54 (2011) 861–874; (in Chinese)http://en.cnki.com.cn/Article_en/CJFDTotal-SXXB201105015.htm.Google Scholar
Woeginger, G. J., “The toughness of split graphs”, Discrete Math. 190 (1998) 295–297; doi:10.1016/S0012-365X(98)00156-3.Google Scholar
Zhang, S. G., Li, X. L. and Han, X. L., “Computing the scattering number of graphs”, Int. J. Comput. Math. 79 (2002) 179–187; doi:10.1080/00207160211919.CrossRefGoogle Scholar
Zhang, S. G., Zhang, Q. L. and Yang, H. W., “Vulnerability parameters of split graphs”, Int. J. Comput. Math. 85 (2008) 19–23; doi:10.1080/00207160701365721.Google Scholar