Abstract
Let \(G = (V,E)\) be a connected graph. The conditional edge connectivity \(\lambda _\delta ^k(G)\) is the cardinality of the minimum edge cuts, if any, whose deletion disconnects \(G\) and each component of \(G - F\) has \(\delta \ge k\). We assume that \(F \subseteq E\) is an edge set, \(F\) is called edge extra-cut, if \(G - F\) is not connected and each component of \(G - F\) has more than \(k\) vertices. The edge extraconnectivity \(\lambda _\mathrm{e}^k(G)\) is the cardinality of the minimum edge extra-cuts. In this paper, we study the conditional edge connectivity and edge extraconnectivity of hypercubes and folded hypercubes.
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References
Bondy JA, Murty USR (1976) Graph theory and its application. Academic Press, New York
Cheng E, Lesniak L, Lipman MJ, Lipták L (2009) Conditional matching preclusion sets. Inf Sci 179:1092–1101
Duh DR, Chen GH, Fang JF (1995) Algorithms and properties of a new two-level network with folded hypercubes as basic modules. IEEE Trans Parallel Distrib Syst 6(7):714–723
Esfahanian AH, Hakimi SL (1988) On computing a conditional edge-connectivity of a graph. Inf Process Lett 27:195–199
El-Amawy A, Latifi S (1991) Properties and performance of folded hypercubes. IEEE Trans Parallel Distrib Syst 2:31–42
F\(\acute{a}\)brega J, Fiol MA (1996) On the extraconnectivity of graphs. Discret Math 155:49–57
Guo L, Guo X (2009) Connectivity of the Mycielskian of a digraph. Appl Math Lett 22:1622–1625
Guo L, Qin C, Guo X (2010) Super connectivity of Kronecker products of graphs. Inf Process Lett 110:659–661
Guo L, Liu R, Guo X (2012) Super connectivity and super edge connectivity of the Mycielskian of a graph. Graph Comb 28:143–147
Harary F (1983) Conditional connectivity. Networks 13:346–357
Liaw SC, Chang GJ (1998) Generalized diameters and Rabin numbers of networks. J Combin Optim 2:371–384
Lai CN, Chen GH, Duh DR (2002) Constructing one-to-many disjoint paths in folded hypercubes. IEEE Trans Comput 51(1):33–45
Latifi S, Hedge M, Naraghi-Pour M (1994) Conditional connectivity measures for large multiprocessor systems. IEEE Trans Comput 43(2):218–222
Lin M-S, Chang M-S, Chen D-J (1999) Efficient algorithms for reliability analysis of distributed computing systems. Inf Sci 117:89–106
Meng JX, Ji YH (2002) On a kind of restricted edge connectivty of graphs. Discret Appl Math 117:183–193
Sim\(\acute{o}\) E, Yebra JLA (1997) The vulnerability of the diameter of folded n-cubes, Discrete Math 174:317–322
Valafar H, Arabnia HR, Williams G (2004) Distributed global optimization and its development on the multiring network. Int J Neural Parallel Sci Comput 12(4):465–490
Wang D (2001) Embedding Hamiltonian cycles into folded hypercubes with faulty links. J Parallel Distrib Comput 61:545–564
Xu JM (2000) Restricted edge connectivity of vertex transitive graphs. Chin Ann Math Ser A 21:605–608
Zhu Q, Xu JM (2006) On restricted edge connectivity and extra edge connectivity of hypercubes and foled hypercubes. J Univ Sci Technol China 36(3):246–253
Zhu Q, Xu JM, Hou X, Xu M (2007) On reliability of the foled hypercubes. Inf Sci 177:1782–1788
Zhang Z, Yuan JJ (2007) Degree conditions for retricted edge connectivity and isoperimetric-edge-connectivity to be optimal. Discret Math. 307:293–298
Acknowledgments
The project is supported by NSFC (No. 11301440, 11171279), Xiamen University of Technology (YKJ12030R), the Foundation to the Educational Committee of Fujian (JA13240, JA13025, JA13034), the Natural Science Foundation of Fujian Province (2013J05006), Natural Sciences Foundation of Guangxi Province (No. 2012GXNSFBA053005, 2011GXNSFA018144). We would like to thank the referees for kind help and valuable suggestions.
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Guo, L., Guo, X. Fault tolerance of hypercubes and folded hypercubes. J Supercomput 68, 1235–1240 (2014). https://doi.org/10.1007/s11227-013-1078-5
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DOI: https://doi.org/10.1007/s11227-013-1078-5