Abstract
In network design, the all-terminal reliability maximization is of paramount importance. In this classical setting, we assume a simple graph with perfect nodes but independent link failures with identical probability \(\rho \). The goal is to communicate p terminals using q links, in such a way that the connectedness probability of the resulting random graph is maximum. A graph with p nodes and q links that meets the maximum reliability property is called uniformly most-reliable (p, q)-graph (UMRG). The discovery of these graphs is a challenging problem that involves an interplay between extremal graph theory and computational optimization. Recent works confirm the existence of special cubic UMRG, such as Wagner, Petersen and Yutsis graphs. To the best of our knowledge, there is no prior works from the literature that find 4-regular UMRG. In this paper, we revisit the breakthroughs in the theory of UMRG. Finally, we mathematically prove that the complement of a cycle with seven nodes, \(\overline{C_7}\), is a 4-regular UMRG. This graph is also identified as an odd antihole using terms from perfect graph theory.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bauer, D., Boesch, F., Suffel, C., Van Slyke, R.: On the validity of a reduction of reliable network design to a graph extremal problem. IEEE Trans. Circuits Syst. 34(12), 1579–1581 (1987)
Bauer, D., Boesch, F., Suffel, C., Tindell, R.: Combinatorial optimization problems in the analysis and design of probabilistic networks. Networks 15(2), 257–271 (1985)
Biggs, N.: Algebraic Graph Theory. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1993)
Boesch, F.T.: On unreliability polynomials and graph connectivity in reliable network synthesis. J. Graph Theory 10(3), 339–352 (1986)
Boesch, F.T., Li, X., Suffel, C.: On the existence of uniformly optimally reliable networks. Networks 21(2), 181–194 (1991)
Boesch, F.T., Satyanarayana, A., Suffel, C.L.: A survey of some network reliability analysis and synthesis results. Networks 54(2), 99–107 (2009)
Bollobás, B.: Extremal Graph Theory. Dover Books on Mathematics, Dover Publications (2004)
Bourel, M., Canale, E., Robledo, F., Romero, P., Stábile, L.: A Hybrid GRASP/VND heuristic for the design of highly reliable networks. In: Blesa Aguilera, M.J., Blum, C., Gambini Santos, H., Pinacho-Davidson, P., Godoy del Campo, J. (eds.) HM 2019. LNCS, vol. 11299, pp. 78–92. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-05983-5_6
Canale, E., Robledo, F., Romero, P., Viera, J.: Building reliability-improving network transformations. In: 2019 15th International Conference on Design of Reliable Communication Networks (DRCN 2019), Coimbra, Portugal, March 2019
Cheng, C.-S.: Maximizing the total number of spanning trees in a graph: two related problems in graph theory and optimum design theory. J. Comb. Theory Ser. B 31(2), 240–248 (1981)
Colbourn, C.J.: The Combinatorics of Network Reliability. Oxford University Press Inc., New York (1987)
Harary, F.: The maximum connectivity of a graph. Proc. Nat. Acad. Sci. U.S.A. 48(7), 1142–1146 (1962)
Kelmans, A.: On graphs with the maximum number of spanning trees. Random Struct. Algorithms 9(1–2), 177–192 (1996)
Kelmans, A.K., Chelnokov, V.M.: A certain polynomial of a graph and graphs with an extremal number of trees. J. Comb. Theory Ser. B 16(3), 197–214 (1974)
Kirchhoff, G.: Über die auflösung der gleichungen, auf welche man bei der untersuchung der linearen verteilung galvanischer ströme geführt wird. Ann. Phys. Chem. 72, 497–508 (1847)
Knuth, D.E.: The art of computer programming: introduction to combinatiorial algorithms and Boolean functions. In: Addison-Wesley Series in Computer Science and Information Proceedings, Addison-Wesley (2008)
Leggett, J.D., Bedrosian, S.D.: On networks with the maximum numbers of trees. In: Proceedings of Eighth Midwest Symposium on Circuit Theory, pp. 1–8, June 1965
Myrvold, W.: Reliable network synthesis: some recent developments. In: Proceedings of the Eighth International Conference on Graph Theory, Combinatorics, Algorithms and Applications, vol. II, pp. 650–660 (1996)
Myrvold, W., Cheung, K.H., Page, L.B., Perry, J.E.: Uniformly-most reliable networks do not always exist. Networks 21(4), 417–419 (1991)
Petingi, L., Boesch, F., Suffel, C.: On the characterization of graphs with maximum number of spanning trees. Discrete Math. 179(1), 155–166 (1998)
Rela, G., Robledo, F., Romero, P.: Petersen graph is uniformly most-reliable. In: Nicosia, G., Pardalos, P., Giuffrida, G., Umeton, R. (eds.) MOD 2017. LNCS, vol. 10710, pp. 426–435. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-72926-8_35
Wang, G.: A proof of Boesch’s conjecture. Networks 24(5), 277–284 (1994)
Acknowledgements
This work is partially supported by Projects 395 CSIC I+D Sistemas Binarios Estocásticos Dinámicos, COST Action 15127 Resilient communication services protecting end-user applications from disaster-based failures, Math-AMSUD Raredep Rare events analysis in multi-component systems with dependent components and STIC-AMSUD ACCON, Algorithms for the capacity crunch problem in optical networks.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Rela, G., Robledo, F., Romero, P. (2019). Uniformly Most-Reliable Graphs and Antiholes. In: Nicosia, G., Pardalos, P., Umeton, R., Giuffrida, G., Sciacca, V. (eds) Machine Learning, Optimization, and Data Science. LOD 2019. Lecture Notes in Computer Science(), vol 11943. Springer, Cham. https://doi.org/10.1007/978-3-030-37599-7_36
Download citation
DOI: https://doi.org/10.1007/978-3-030-37599-7_36
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-37598-0
Online ISBN: 978-3-030-37599-7
eBook Packages: Computer ScienceComputer Science (R0)