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Connectivity in Hypergraphs

Published online by Cambridge University Press:  20 November 2018

Megan Dewar ,
David Pike  and
John Proos
Megan Dewar
Affiliation:
Tutte Institute for Mathematics and Computing, Ottawa, ON, e-mail: tutte.institute+megandewar@gmail.com
David Pike
Affiliation:
Tutte Institute for Mathematics and Computing, Ottawa, ON, e-mail: tutte.institute+johnproos@gmail.com
John Proos
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, e-mail: dapike@mun.ca
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Abstract

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In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that, while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard when restricted to hypergraphs with maximum edge size at most 3. We also discuss the relationship between strong vertex connectivity and the minimum transversal problem for hypergraphs, showing that there are classes of hypergraphs for which one of the problems is NP-hard, while the other can be solved in polynomial time.

Keywords

05C6505C4068Q17hypergraphconnectivitycomputational complexitytransversal

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 2 , 01 June 2018 , pp. 252 - 271
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Bahmanian, M. A. and Sajna, M., Connection and Separation in hypergraphs. Theory Appl. Graphs 2 (2015), no. 2, Art. 5, 24.Google Scholar
[2] Bäräsz, M., Becker, J., and Frank, A., An algorithm for source location in directed graphs. Oper. Res. Lett. 33 (2005), 221–230. http://dx.doi.Org/10.1016/j.orl.2004.07.005Google Scholar
[3] Bondy, J. A. and Murty, U. S. R., Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008.Google Scholar
[4] Bretto, A., Hypergraph theory: An introduction. Mathematical Engineering, Springer, Cham, 2013.Google Scholar
[5] Chekuri, C. and Xu, C., Computing minimum cuts in hypergraphs. In: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (Barcelona, 2017), SIAM, Philadelphia, PA, 2017, pp. 1085–1100.Google Scholar
[6] Cheng, E., Edge-augmentation of hypergraphs. Connectivity augmentation ofnetworks: structures and algorithms (Budapest, 1994). Math. Program. 84(1999), no. 3, Ser. B, 443–465. http://dx.doi.Org/10.1007/s101070050032Google Scholar
[7] Duchet, P., Hypergraphs. In: Handbook of combinatorics, Vol. 1, Elsevier Sei. B. V., Amsterdam, 1995, pp. 381–432.Google Scholar
[8] Edmonds, J. and Karp, R. M., Theoretical improvements in algorithmic efficiencyfor networkflow Problems. In: Combinatorial Structure and their Applications, Proc. Calgary Internat. Conf., Calgarym Alta., Gordon and Breach, New York, 1970, pp. 93–96.Google Scholar
[9] Frank, A., Edge-connection of graphs, digraphs, and hypergraphs. In: More sets, graphs and numbers, Bolyai Soc. Math. Stud., 15, Springer, Berlin, 2006, pp. 93–141.Google Scholar
[10] Karp, R. M., Reducibility among combinatorial problems. In: Complexity of Computer computations, Plenum Press, New York, 1972, pp. 85–103.Google Scholar
[11] Nagamochi, H. and Ibaraki, T., Algorithmic aspects ofgraph Connectivity. Encyclopedia of Mathematics and its Applications, 123, Cambridge University Press, Cambridge, 2008.Google Scholar
[12] Slater, P. J., A characterization ofsoft hypergraphs. Canad. Math. Bull. 21 (1978), 335–337. http://dx.doi.Org/10.4153/CMB-1978-058-5Google Scholar
[13] Voloshin, V. I.. Introduction to graph and hypergraph theory. Nova Science Publishers, Ine, New York, 2009.Google Scholar
[14] Whitney, H., Congruent graphs and the Connectivity of graphs. Amer. J. Math. 54 (1932), 150–168. http://dx.doi.Org/10.2307/2371086Google Scholar