Abstract
In designing and understanding of computer networks, how to improve network robustness or protect a network from vulnerability remains an overarching concern. The residual closeness is a measure of network vulnerability and robustness even when the removal of vertices does not disconnect the underlying graph. We determine all the graphs that minimize and maximize the residual closeness respectively over all n-vertex connected graphs with r cut vertices, where \(1\le r\le n-3\).
Similar content being viewed by others
Data availibility
Not applicable.
References
Aytac A, Berberler ZNO (2017) Robustness of regular caterpillars. Internat J Found Comput Sci 28(7):835–841. https://doi.org/10.1142/S0129054117500277
Aytac A, Odabas ZN (2011) Residual closeness of wheels and related networks. Internat J Found Comput Sci 22(5):1229–1240. https://doi.org/10.1142/S0129054111008660
Aytac A, Odabas Berberler ZN (2018) Network robustness and residual closeness. RAIRO Oper Res 52(3):839–847. https://doi.org/10.1051/ro/2016071
Boccaletti S, Buldú J, Criado R, Flores J, Latora V, Pello J (2007) Multiscale vulnerability of complex networks. Chaos 17(4):043110. https://doi.org/10.1063/1.2801687
Cheng M, Zhou B (2022) Residual closeness of graphs with given parameters. J Oper Res Soc China. https://doi.org/10.1007/s40305-022-00405-9
Chvátal V (1973) Tough graphs and Hamiltonian circuits. Discrete Math 5:215–228. https://doi.org/10.1016/0012-365X(73)90138-6
Dangalchev C (2006) Residual closeness in networks. Phys A 365(2):556–564. https://doi.org/10.1016/j.physa.2005.12.020
Dangalchev C (2011) Residual closeness and generalized closeness. Internat J Found Comput Sci 22(8):1939–1948. https://doi.org/10.1142/S0129054111009136
Dangalchev C (2018) Residual closeness of generalized thorn graphs. Fund Inform 162(1):1–15
Frank H, Frisch IT (1970) Analysis and design of survivable networks. IEEE Trans Commun Tech 18:501–519
Holme P, Kim BJ, Yoon CN, Han SK (2002) Attack vulnerability of complex networks. Phys Rev E 65(5):056109
Jackson MO (2008) Social and Economic Networks. Princeton University Press, Princeton, New Jersey
Jung HA (1978) On a class of posets and the corresponding comparability graphs. J Combin Theory Ser B 24:125–133. https://doi.org/10.1016/0095-8956(78)90013-8
Odabas ZN, Aytac A (2013) Residual closeness in cycles and related networks. Fund Inform 124(3):297–307
Turaci T, Ökten M (2015) Vulnerability of Mycielski graphs via residual closeness. Ars Combin 118:419–427
Wang Y, Zhou B (2022) Residual closeness, matching number and chromatic number. Comput J. https://doi.org/10.1093/comjnl/bxac004
Zhou B, Li Z, Guo H (2021) Extremal results on vertex and link residual closeness. Internat J Found Comput Sci 32(8):921–941. https://doi.org/10.1142/S0129054121500295
Acknowledgements
The authors thank the referees for helpful comments.
Funding
This work was supported by the National Natural Science Foundation of China (No. 12071158).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest
The authors declare that they have no conflict of interest.
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
Li, C., Xu, L. & Zhou, B. On the residual closeness of graphs with cut vertices. J Comb Optim 45, 115 (2023). https://doi.org/10.1007/s10878-023-01042-5
Accepted:
Published:
DOI: https://doi.org/10.1007/s10878-023-01042-5