Abstract
For a connected graph G, a set S of vertices is a cyclic vertex cutset if \(G - S\) is not connected and at least two components of \(G-S\) contain a cycle respectively. The cyclic vertex connectivity \(c \kappa (G)\) is the cardinality of a minimum cyclic vertex cutset. In this paper, for a k-regular graph G with fixed k value, we give a polynomial time algorithm to determine \(c \kappa (G)\) and its time complexity is bounded by \(O(v^{15/2})\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Dinitz (2006) tells the differences between his version and Even’s version.
References
Bondy JA, Murty USR (1976) Graph theory with applications. MacMillan Press, London
Dinitz Y (2006) Dinitz’ algorithm: the original version and Even’s version, Theoretical Computer Science. Springer, Berlin, pp 218–240
Dvořák Z, Kára J, král’ D, Pangrác O (2004) An algorithm for cyclic edge connectivity of cubic graphs. In: SWAT 2004, LNCS 3111, pp 236–247
Even S (2011) Graph algorithms. Cambridge University Press, Cambridge
Even S, Tarjan RE (1975) Network flow and testing graph connectivity. SIAM J Comput 4:507–518
Liang J, Lou D, Qin Z (2016a) A polynomial algorithm determining cyclic vertex connectivity of 4-regular graphs, Manuscript
Liang J, Lou D, Zhang Z (2016b) A polynomial time algorithm for cyclic vertex connectivity of cubic graphs. J Comput Math Int. https://doi.org/10.1080/00207160.2016.1210792
Lou D, Liang K (2014) An improved algorithm for cyclic edge connectivity of regular graphs. Ars Comb 115:315–333
Lou D, Wang W (2005) An efficient algorithm for cyclic edge connectivity of regular graphs. Ars Comb 77:311–318
McCuaig WD (1992) Edge reductions in cyclically k-connected cubic graphs. J Comb Theory Ser B 56:16–44
Nedela R, Škoviera M (1995) Atoms of cyclic connectivity in cubic graphs. Math Slovaca 45:481–499
Nedela R, Škoviera M (1996) Decompositions and reductions of snarks. J Graph Theory 22:253–279
Peroche B (1983) On several sorts of connectivity. Discrete Math 46:267–277
Tait PG (1880) Remarks on the coloring of maps. Proc R Soc Edinb 10:501–503
Tutte WT (1960) A non-Hamiltonian planar graph. Acta Math Acad Sci Hung 11:371–375
Acknowledgements
This work was supported by The Ph.D. Start-up Fund of Natural Science Foundation of Guangdong Province (Grant No. 2018A030310516), The Creative Talents Project Fund of Guangdong Province Department of Education (Natural Science) (Grant No. 2017KQNCX053) and Guangdong Natural Science Foundation (Grant No. 2016A030313829).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liang, J., Lou, D. A polynomial algorithm determining cyclic vertex connectivity of k-regular graphs with fixed k. J Comb Optim 37, 1000–1010 (2019). https://doi.org/10.1007/s10878-018-0332-4
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-018-0332-4