Abstract
We relate the number of minimum cuts in a weighted undirected graph with various structural parameters of the graph. In particular, we upper—bound the number of minimum cuts in terms of the radius, diameter, minimum degree, maximum degree, chordality, expansion, girth etc. of the graph.
This research is supported in part by the Infosys Fellowship.
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R.E. Bixby, The Minimum Number of Edges and Vertices in a Graph with Edge Connectivity n and m n-bonds. Networks, Vol.5, (1975) 253–298.
F.T. Boesch, Synthesis Of Reliable Networks — A Survey. IEEE Transactions On Reliability, vol R-35, (1986) 240–246.
B. Bollabos, Extremal Graph Theory. Academic Press, London, 1978.
M.O. Ball, J.S. Provan, Calculating bounds on reachability and connectedness in stochastic networks. Networks, vol 5, (1975) 253–298.
M.O. Ball, J.S. Provan, The Complexity Of Counting Cuts And Of Computing The Probability That A Graph Is Connected. SIAM Journal of Computing, 12, (1983) 777–788.
M.O. Ball, J.S. Provan, Computing Network Reliability In Time Polynomial In The Number Of Cuts. Operations Research, 32, (1984) 516–521.
E.A. Dinits, A.V. Karzanov, M.V. Lomosonov, On the Structure of a Family of Minimal Weighted Cuts in a Graph. Studies in Discrete Optimization [In Russian], A.A. Friedman (Ed), Nauka, Moscow (1976) 290–306.
H.N. Gabow, A Matroid Approach To Finding Edge Connectivity And Packing Arborescences. Proceedings Of 23rd Annual ACM-SIAM Symposium On Theory Of Computing, (1991) 112–122.
F. Harary, Graph Theory. Addison-Wesley Reading, MA, 1969.
Lisa Fleischer, Building Chain And Cactus Representations Of All Minimum Cuts From Hao—Orlin In The Same Asymptotic Run Time. Journal Of Algorithms, 33, (1999) 51–72.
D.R. Karger, Random Sampling In Cut, Flow and Network Design Problems. In Proceedings Of 6th Annual ACM-SIAM Symposium On Discrete Algorithms, (1995), 648–657.
Jeno Lehel, Frederic Maffray, Myriam Preissmann, Graphs With Largest Number Of Minimum Cuts. Discrete Applied Mathematics, 65, (1996) 387–407.
A. Kanevsky, Graphs With Odd And Even Edge Connectivity Are Inherently Different. Tech. report, TAMU-89-10, June 1989.
H. Nagamochi, K. Nishimura, T. Ibaraki, Computing All Small Cuts In An Undirected Network. SIAM Journal Of Discrete Math, 10(3), (1997) 469–481.
D. Naor, Vijay. V. Vazirani, Representing and Enumerating Edge Connectivity Cuts in RNC Workshop On Algorithms and Data structures (1991) LNCS 519 273–285.
J.S. Provan, Bounds On The Reliability Of Networks. IEEE Transactions On Reliability, R-35, (1986) 26–268.
J.C. Picard, M. Queyranne, On The Structure Of All Minimum Cuts In A Network And Applications. Mathematical programming Study, 13, (1980) 8–16.
V.V. Vazirani and M. Yannakakis, Suboptimal Cuts: Their Enumeration, Weight, And Number. Lecture Notes in Computer Science, 623, Springer-Verlag, (1992) 366–377.
M.R. Henzinger, D.P. Williamson, On The Number Of Small Cuts. Information Processing Letters, 59, (1996), 41–44.
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Chandran, L.S., Ram, L.S. (2002). On the Number of Minimum Cuts in a Graph. In: Ibarra, O.H., Zhang, L. (eds) Computing and Combinatorics. COCOON 2002. Lecture Notes in Computer Science, vol 2387. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45655-4_25
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DOI: https://doi.org/10.1007/3-540-45655-4_25
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