Abstract
For a connected graph G, a vertex subset F ⊂ V(G) is a cyclic vertex-cut of G if G − F is disconnected and at least two of its components contain cycles. The cardinality of a minimum cyclic vertex-cut of G, denoted by κ c (G), is the cyclic vertex-connectivity of G. In this paper, we show that for any integer n ≥ 4, the n-dimensional star graph SG n has κ c (SG n ) = 6(n − 3).
This research is supported by NSFC (10971255), the Key Project of Chinese Ministry of Education (208161), Program for New Century Excellent Talents in University, and The Project-sponsored by SRF for ROCS, SEM.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Akers, S.B., Harel, D., Krishnamurthy, B.: The star graph: an attractive alternative to the n-cube. In: Proc. Int. Conf. Parallel Processing, pp. 393–400 (1987)
Akers, S.B., Krishnamurthy, B.: A group-theoretic model for symmetric interconnection networks. IEEE Transactions on Computers 38, 555–566 (1989)
Bondy, J.A., Murty, U.S.R.: Graph theory with application. Macmillan, London (1976)
Cheng, E., Lipman, M.J.: Increasing the connectivity of the star graphs. Networks 40, 165–169 (2002)
Day, K., Tripathi, A.: A comparative study of topological properties of hypercubes and star graphs. IEEE Trans. Comp. 5, 31–38 (1994)
Esfahanian, A.H.: Generalized measures of fault tolerance with application to n-cube networks. IEEE Trans. Comp. 38, 1586–1591 (1989)
Harary, F.: Conditional connectivity. Networks 13, 347–357 (1983)
Heydemann, M.C., Ducourthial, B.: Cayley graphs and interconnection networks. In: Hahn, G., Sabidussi, G. (eds.) Graph Symmetry, Montreal, PQ. NATO Advanced Science Institutes Series C, Mathematica and Physical Sciences, vol. 497, pp. 167–224. Kluwer Academic Publishers, Dordrecht (1996)
Holton, D.A., Lou, D., Plummer, M.D.: On the 2-extendability of plannar graphs. Discrete Math. 96, 81–99 (1991)
Hu, S.C., Yang, C.B.: Fault tolerance on star graphs. In: Proceedings of the First Aizu International Symposium on Parallel Algorithms/Architecture Synthesis, pp. 176–182 (1995)
Latifi, S., Hegde, M., Pour, M.N.: Conditional connectivity measures for large multiprocessor systems. IEEE Trans. Comp. 43, 218–222 (1994)
Lou, D., Holton, D.A.: Lower bound of cyclic edge connectivity for n-extendability of regular graphs. Discrete Math. 112, 139–150 (1993)
Nedela, R., Skoviera, M.: Atoms of cyclic connectivity in cubic graphs. Math. Slovaca 45, 481–499 (1995)
Plummer, M.D.: On the cyclic connectivity of planar graphs. Lecture Notes in Mathematics, vol. 303, pp. 235–242 (1972)
Robertson, N.: Minimal cyclic-4-connected graphs. Trans. Amer. Math. Soc. 284, 665–684 (1984)
Tait, P.G.: Remarks on the colouring of maps. Proc. Roy. Soc., Edinburgh 10, 501–503 (1880)
Wan, M., Zhang, Z.: A kind of conditional vertex connectivity of star graphs. Appl. Math. Letters 22, 264–267 (2009)
Wang, B., Zhang, Z.: On cyclic edge-connectivity of transitive graphs. Discrete Math. 309, 4555–4563 (2009)
Watkins, M.E.: Connectivity of transitive graphs. J. Combin. Theory 8, 23–29 (1970)
Xu, J.M., Liu, Q.: 2-restricted edge connectivity of vertex-transitive graphs. Australasian Journal of Combinatorics 30, 41–49 (2004)
Zhang, C.Q.: Integer flows and cycle covers of graphs. Marcel Dekker Inc., New York (1997)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yu, Z., Liu, Q., Zhang, Z. (2010). Cyclic Vertex Connectivity of Star Graphs. In: Wu, W., Daescu, O. (eds) Combinatorial Optimization and Applications. COCOA 2010. Lecture Notes in Computer Science, vol 6508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17458-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-17458-2_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17457-5
Online ISBN: 978-3-642-17458-2
eBook Packages: Computer ScienceComputer Science (R0)