On the Role of the High-Low Partition in Realizing a Degree Sequence by a Bipartite Graph

Authors Amotz Bar-Noy, Toni Böhnlein, David Peleg, Dror Rawitz

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Amotz Bar-Noy
  • City University of New York (CUNY), NY, USA
Toni Böhnlein
  • Weizmann Institute of Science, Rehovot, Israel
David Peleg
  • Weizmann Institute of Science, Rehovot, Israel
Dror Rawitz
  • Bar Ilan University, Ramat-Gan, Israel

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Amotz Bar-Noy, Toni Böhnlein, David Peleg, and Dror Rawitz. On the Role of the High-Low Partition in Realizing a Degree Sequence by a Bipartite Graph. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We consider the problem of characterizing degree sequences that can be realized by a bipartite graph. If a partition of the sequence into the two sides of the bipartite graph is given as part of the input, then a complete characterization has been established over 60 years ago. However, the general question, in which a partition and a realizing graph need to be determined, is still open. We investigate the role of an important class of special partitions, called High-Low partitions, which separate the degrees of a sequence into two groups, the high degrees and the low degrees. We show that when the High-Low partition exists and satisfies some natural properties, analysing the High-Low partition resolves the bigraphic realization problem. For sequences that are known to be not realizable by a bipartite graph or that are undecided, we provide approximate realizations based on the High-Low partition.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Graph Realization
  • Bipartite Graphs
  • Degree Sequences
  • Graphic Sequences
  • Bigraphic Sequences
  • Approximate Realization
  • Multigraph Realization


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  1. P. Adams and Y. Nikolayevsky. Planar bipartite biregular degree sequences. Discrete Mathematics, 342:433-440, 2019. Google Scholar
  2. G. E Andrews. The Theory of Partitions. Cambridge University Press, 1998. Google Scholar
  3. C. Avin, M. Borokhovich, Z. Lotker, and D. Peleg. Distributed computing on core-periphery networks: Axiom-based design. Journal of Parallel and Distributed Computing, 99:51-67, 2017. Google Scholar
  4. C. Avin, Z. Lotker, Y. Nahum, and D. Peleg. Core size and densification in preferential attachment networks. In International Colloquium on Automata, Languages, and Programming, pages 492-503. Springer, 2015. Google Scholar
  5. C. Avin, Z. Lotker, D. Peleg, Y.-A. Pignolet, and I. Turkel. Elites in social networks: An axiomatic approach to power balance and price’s square root law. PloS one, 13(10):e0205820, 2018. Google Scholar
  6. A. Bar-Noy, T. Böhnlein, D. Peleg, M. Perry, and D. Rawitz. Relaxed and approximate graph realizations. In International Workshop on Combinatorial Algorithms, pages 3-19. Springer, 2021. Google Scholar
  7. A. Bar-Noy, T. Böhnlein, D. Peleg, and D. Rawitz. On Realizing a Single Degree Sequence by a Bipartite Graph. In 18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022), volume 227, pages 1:1-1:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  8. A. Bar-Noy, T. Böhnlein, D. Peleg, and D. Rawitz. On realizing even degree sequences by bipartite graphs. Unpublished manuscript, 2022. Google Scholar
  9. R. Bellman. Notes on the theory of dynamic programming iv-maximization over discrete sets. Naval Research Logistics Quarterly, 3(1-2):67-70, 1956. Google Scholar
  10. J. K. Blitzstein and P. Diaconis. A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Mathematics, 6(4):489-522, 2011. Google Scholar
  11. S. P Borgatti and M. G Everett. Models of core/periphery structures. Social Networks, 21(4):375-395, 2000. Google Scholar
  12. A. A. Chernyak, Z. A. Chernyak, and R. I. Tyshkevich. On forcibly hereditary p-graphical sequences. Discrete Mathematics, 64:111-128, 1987. Google Scholar
  13. S. A. Choudum. A simple proof of the Erdös-Gallai theorem on graph sequences. Bull. Austral. Math. Soc., 33(1):67-70, 1991. Google Scholar
  14. V Chungphaisan. Conditions for sequences to be r-graphic. Discrete Mathematics, 7(1-2):31-39, 1974. Google Scholar
  15. B. Cloteaux. Fast sequential creation of random realizations of degree sequences. Internet Mathematics, 12(3):205-219, 2016. Google Scholar
  16. T. H Cormen, C. E Leiserson, R. L Rivest, and C. Stein. Introduction to algorithms. MIT press, 2009. Google Scholar
  17. Dóra Erdös, R. Gemulla, and E. Terzi. Reconstructing graphs from neighborhood data. ACM Trans. Knowledge Discovery from Data, 8(4):23:1-23:22, 2014. Google Scholar
  18. P. Erdös and T. Gallai. Graphs with prescribed degrees of vertices [hungarian]. Matematikai Lapok, 11:264-274, 1960. Google Scholar
  19. D. Gale. A theorem on flows in networks. Pacific J. Math., 7:1073-1082, 1957. Google Scholar
  20. G. Gupta, P. Joshi, and A. Tripathi. Graphic sequences of trees and a problem of Frobenius. Czechoslovak Math. J., 57:49-52, 2007. Google Scholar
  21. S. Louis Hakimi. On realizability of a set of integers as degrees of the vertices of a linear graph -I. SIAM J. Appl. Math., 10(3):496-506, 1962. Google Scholar
  22. P. L. Hammer, T. Ibaraki, and B. Simeone. Threshold sequences. SIAM J. Algebra. Discr., 2(1):39-49, 1981. Google Scholar
  23. P. L. Hammer and B. Simeone. The splittance of a graph. Combinatorica, 1:275-284, 1981. Google Scholar
  24. V. Havel. A remark on the existence of finite graphs [in Czech]. Casopis Pest. Mat., 80:477-480, 1955. Google Scholar
  25. H. Hulett, T. G Will, and G. J Woeginger. Multigraph realizations of degree sequences: Maximization is easy, minimization is hard. Operations Research Letters, 36(5):594-596, 2008. Google Scholar
  26. P.J. Kelly. A congruence theorem for trees. Pacific J. Math., 7:961-968, 1957. Google Scholar
  27. D. J. Kleitman. Minimal number of multiple edges in realization of an incidence sequence without loops. SIAM Journal on Applied Mathematics, 18(1):25-28, 1970. Google Scholar
  28. P. Marchioro, A. Morgana, R. Petreschi, and B. Simeone. Degree sequences of matrogenic graphs. Discrete Mathematics, 51(1):47-61, 1984. URL: https://doi.org/10.1016/0012-365X(84)90023-2.
  29. M. Mihail and N. Vishnoi. On generating graphs with prescribed degree sequences for complex network modeling applications. 3rd ARACNE, 2002. Google Scholar
  30. J. W Miller. Reduced criteria for degree sequences. Discrete Mathematics, 313(4):550-562, 2013. Google Scholar
  31. A. B Owens. On determining the minimum number of multiple edges for an incidence sequence. SIAM J. on Applied Math., 18(1):238-240, 1970. Google Scholar
  32. AB Owens and HM Trent. On determining minimal singularities for the realizations of an incidence sequence. SIAM J. on Applied Math., 15(2):406-418, 1967. Google Scholar
  33. S. B. Rao. A survey of the theory of potentially p-graphic and forcibly p-graphic degree sequences. In Combinatorics and graph theory, volume 885 of LNM, pages 417-440, 1981. Google Scholar
  34. M P. Rombach, M. A Porter, J. H Fowler, and P. J Mucha. Core-periphery structure in networks. SIAM Journal on Applied mathematics, 74(1):167-190, 2014. Google Scholar
  35. H.J. Ryser. Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9:371-377, 1957. Google Scholar
  36. E. F. Schmeichel and S. L. Hakimi. On planar graphical degree sequences. SIAM J. Applied Math., 32:598-609, 1977. Google Scholar
  37. G. Sierksma and H. Hoogeveen. Seven criteria for integer sequences being graphic. J. Graph Theory, 15(2):223-231, 1991. Google Scholar
  38. A. Tatsuya and H. Nagamochi. Comparison and enumeration of chemical graphs. Computational and structural biotechnology, 5, 2013. Google Scholar
  39. A. Tripathi and H. Tyagi. A simple criterion on degree sequences of graphs. Discrete Applied Mathematics, 156(18):3513-3517, 2008. Google Scholar
  40. R. Tyshkevich. Decomposition of graphical sequences and unigraphs. Discrete Mathematics, 220:201-238, 2000. Google Scholar
  41. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: A survey, I. Cybernetics, 23:734-745, 1987. Google Scholar
  42. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: A survey, II. Cybernetics, 24:137-152, 1988. Google Scholar
  43. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: A survey, III. Cybernetics, 24:539-548, 1988. Google Scholar
  44. S.M. Ulam. A collection of mathematical problems. Wiley, 1960. Google Scholar
  45. N.C. Wormald. Models of random regular graphs. Surveys in Combin., 267:239-298, 1999. Google Scholar
  46. X. Zhang, T. Martin, and M. EJ Newman. Identification of core-periphery structure in networks. Physical Review E, 91(3):032803, 2015. Google Scholar
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