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

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## Cite As

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)
https://doi.org/10.4230/LIPIcs.MFCS.2022.14

## Abstract

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
##### Keywords
• Graph Realization
• Bipartite Graphs
• Degree Sequences
• Graphic Sequences
• Bigraphic Sequences
• Approximate Realization
• Multigraph Realization

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## References

1. P. Adams and Y. Nikolayevsky. Planar bipartite biregular degree sequences. Discrete Mathematics, 342:433-440, 2019.
2. G. E Andrews. The Theory of Partitions. Cambridge University Press, 1998.
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.
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.
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.
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.
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.
8. A. Bar-Noy, T. Böhnlein, D. Peleg, and D. Rawitz. On realizing even degree sequences by bipartite graphs. Unpublished manuscript, 2022.
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.
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.
11. S. P Borgatti and M. G Everett. Models of core/periphery structures. Social Networks, 21(4):375-395, 2000.
12. A. A. Chernyak, Z. A. Chernyak, and R. I. Tyshkevich. On forcibly hereditary p-graphical sequences. Discrete Mathematics, 64:111-128, 1987.
13. S. A. Choudum. A simple proof of the Erdös-Gallai theorem on graph sequences. Bull. Austral. Math. Soc., 33(1):67-70, 1991.
14. V Chungphaisan. Conditions for sequences to be r-graphic. Discrete Mathematics, 7(1-2):31-39, 1974.
15. B. Cloteaux. Fast sequential creation of random realizations of degree sequences. Internet Mathematics, 12(3):205-219, 2016.
16. T. H Cormen, C. E Leiserson, R. L Rivest, and C. Stein. Introduction to algorithms. MIT press, 2009.
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.
18. P. Erdös and T. Gallai. Graphs with prescribed degrees of vertices [hungarian]. Matematikai Lapok, 11:264-274, 1960.
19. D. Gale. A theorem on flows in networks. Pacific J. Math., 7:1073-1082, 1957.
20. G. Gupta, P. Joshi, and A. Tripathi. Graphic sequences of trees and a problem of Frobenius. Czechoslovak Math. J., 57:49-52, 2007.
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.
22. P. L. Hammer, T. Ibaraki, and B. Simeone. Threshold sequences. SIAM J. Algebra. Discr., 2(1):39-49, 1981.
23. P. L. Hammer and B. Simeone. The splittance of a graph. Combinatorica, 1:275-284, 1981.
24. V. Havel. A remark on the existence of finite graphs [in Czech]. Casopis Pest. Mat., 80:477-480, 1955.
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.
26. P.J. Kelly. A congruence theorem for trees. Pacific J. Math., 7:961-968, 1957.
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.
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.
30. J. W Miller. Reduced criteria for degree sequences. Discrete Mathematics, 313(4):550-562, 2013.
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.
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.
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.
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.
35. H.J. Ryser. Combinatorial properties of matrices of zeros and ones. Canad. J. Math., 9:371-377, 1957.
36. E. F. Schmeichel and S. L. Hakimi. On planar graphical degree sequences. SIAM J. Applied Math., 32:598-609, 1977.
37. G. Sierksma and H. Hoogeveen. Seven criteria for integer sequences being graphic. J. Graph Theory, 15(2):223-231, 1991.
38. A. Tatsuya and H. Nagamochi. Comparison and enumeration of chemical graphs. Computational and structural biotechnology, 5, 2013.
39. A. Tripathi and H. Tyagi. A simple criterion on degree sequences of graphs. Discrete Applied Mathematics, 156(18):3513-3517, 2008.
40. R. Tyshkevich. Decomposition of graphical sequences and unigraphs. Discrete Mathematics, 220:201-238, 2000.
41. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: A survey, I. Cybernetics, 23:734-745, 1987.
42. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: A survey, II. Cybernetics, 24:137-152, 1988.
43. R. I. Tyshkevich, A. A. Chernyak, and Z. A. Chernyak. Graphs and degree sequences: A survey, III. Cybernetics, 24:539-548, 1988.
44. S.M. Ulam. A collection of mathematical problems. Wiley, 1960.
45. N.C. Wormald. Models of random regular graphs. Surveys in Combin., 267:239-298, 1999.
46. X. Zhang, T. Martin, and M. EJ Newman. Identification of core-periphery structure in networks. Physical Review E, 91(3):032803, 2015.
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