Document Open Access Logo

Selected Neighbor Degree Forest Realization

Authors Amotz Bar-Noy, David Peleg , Dror Rawitz, Elad Yehezkel

PDF
Thumbnail PDF

File

LIPIcs.ISAAC.2021.27.pdf
  • Filesize: 0.75 MB
  • 15 pages

Document Identifiers

Author Details

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

Funding

Supported in part by a US-Israel BSF grant (2018043).

Cite As Get BibTex

Amotz Bar-Noy, David Peleg, Dror Rawitz, and Elad Yehezkel. Selected Neighbor Degree Forest Realization. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ISAAC.2021.27

Abstract

The classical degree realization problem is defined as follows: Given a sequence d̄ = (d_1,…,d_n) of positive integers, construct an n-vertex graph in which each vertex u_i has degree d_i (or decide that no such graph exists). In this article, we present and study the related selected neighbor degree realization problem, which requires that each vertex u_i of G has a neighbor of degree d_i. We solve the problem when G is required to be acyclic (i.e., a forest), and present a sufficient and necessary condition for a given sequence to be realizable.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • network realization
  • graph algorithms
  • lower bound

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. M. Aigner and E. Triesch. Realizability and uniqueness in graphs. Discrete Math., 136:3-20, 1994. Google Scholar
  2. Takao Asano. Graphical degree sequence problems with connectivity requirements. In 4th ISAAC, volume 762 of LNCS, pages 38-47, 1993. Google Scholar
  3. A. Bar-Noy, K. Choudhary, A. Cohen, D. Peleg, and D. Rawitz. Minimum neighboring degree realization in graphs and trees. In 28th ESA, volume 173 of LIPIcs, pages 10:1-10:15, 2020. Google Scholar
  4. Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Realizability of graph specifications: Characterizations and algorithms. In 25th International Colloquium on Structural Information and Communication Complexity, volume 11085 of LNCS, pages 3-13, 2018. Google Scholar
  5. Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Graph profile realizations and applications to social networks. In 13th WALCOM, volume 11355 of LNCS, pages 3-14, 2019. Google Scholar
  6. Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Graph realizations: Maximum degree in vertex neighborhoods. In 17th SWAT, volume 162 of LIPIcs, pages 10:1-10:17, 2020. Google Scholar
  7. Amotz Bar-Noy, David Peleg, Mor Perry, and Dror Rawitz. Composed degree-distance realizations of graphs. In 32nd IWOCA, volume 12757 of LNCS, pages 63-77, 2021. Google Scholar
  8. Vladimir Batagelj, Tomaz Pisanski, and J. M. S. Simões-Pereira. An algorithm for tree-realizability of distance matrices. Int. J. Comput. Math., 34(3-4):171-176, 1990. Google Scholar
  9. Joseph K. Blitzstein and Persi Diaconis. A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Mathematics, 6(4):489-522, 2011. Google Scholar
  10. W. Chou and H. Frank. Survivable communication networks and the terminal capacity matrix. IEEE Trans. Circuit Theory, CT-17:192-197, 1970. Google Scholar
  11. S.A. Choudum. A simple proof of the Erdös-Gallai theorem on graph sequences. Bulletin of the Australian Mathematical Society, 33:67-70, 1986. Google Scholar
  12. Fan R. K. Chung, Mark W. Garrett, Ronald L. Graham, and David Shallcross. Distance realization problems with applications to internet tomography. J. Comput. Syst. Sci., 63(3):432-448, 2001. Google Scholar
  13. Brian Cloteaux. Fast sequential creation of random realizations of degree sequences. Internet Mathematics, 12(3):205-219, 2016. Google Scholar
  14. Joseph C. Culberson and Piotr Rudnicki. A fast algorithm for constructing trees from distance matrices. Inf. Process. Lett., 30(4):215-220, 1989. Google Scholar
  15. P. Erdös and T. Gallai. Graphs with prescribed degrees of vertices [hungarian]. Matematikai Lapok, 11:264-274, 1960. Google Scholar
  16. 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
  17. S. Louis Hakimi and S. S. Yau. Distance matrix of a graph and its realizability. Quart. Appl. Math., 22:305-317, 1965. Google Scholar
  18. S.L. 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
  19. V. Havel. A remark on the existence of finite graphs [in Czech]. Casopis Pest. Mat., 80:477-480, 1955. Google Scholar
  20. P.J. Kelly. A congruence theorem for trees. Pacific J. Math., 7:961-968, 1957. Google Scholar
  21. Milena Mihail and Nisheeth Vishnoi. On generating graphs with prescribed degree sequences for complex network modeling applications. 3rd Workshop on Approximation and Randomization Algorithms in Communication Networks, 2002. Google Scholar
  22. Elchanan Mossel and Nathan Ross. Shotgun assembly of labeled graphs. CoRR, abs/1504.07682, 2015. URL: http://arxiv.org/abs/1504.07682.
  23. Juhani Nieminen. Realizing the distance matrix of a graph. J. Inf. Process. Cybern., 12(1/2):29-31, 1976. Google Scholar
  24. P.V. O'Neil. Ulam’s conjecture and graph reconstructions. Amer. Math. Monthly, 77:35-43, 1970. Google Scholar
  25. A. N. Patrinos and S. L. Hakimi. The distance matrix of a graph and its tree realizability. Quarterly of Applied Math., 255, 1972. Google Scholar
  26. G. Sierksma and H. Hoogeveen. Seven criteria for integer sequences being graphic. J. Graph Theory, 15:223-231, 1991. Google Scholar
  27. J. M. S. Simões-Pereira. A note on the tree realizability of a distance matrix. J. Combinat. Theory B, 6:303-310, 1969. Google Scholar
  28. J. M. S. Simões-Pereira. A note on distance matrices with unicyclic graph realizations. Discrete Mathematics, 65:277-287, 1987. Google Scholar
  29. J. M. S. Simões-Pereira. An algorithm and its role in the study of optimal graph realizations of distance matrices. Discrete Mathematics, 79(3):299-312, 1990. Google Scholar
  30. Hiroshi Tamura, Masakazu Sengoku, Shoji Shinoda, and Takeo Abe. Realization of a network from the upper and lower bounds of the distances (or capacities) between vertices. In IEEE ISCAS, pages 2545-2548. IEEE, 1993. Google Scholar
  31. A. Tripathi and H. Tyagi. A simple criterion on degree sequences of graphs. Discrete Applied Math., 156:3513-3517, 2008. Google Scholar
  32. S.M. Ulam. A collection of mathematical problems. Wiley, 1960. Google Scholar
  33. Sacha C. Varone. A constructive algorithm for realizing a distance matrix. Eur. J. Oper. Res., 174(1):102-111, 2006. Google Scholar
  34. D.L. Wang and D.J. Kleitman. On the existence of n-connected graphs with prescribed degrees (n > 2). Networks, 3:225-239, 1973. Google Scholar
  35. N.C. Wormald. Models of random regular graphs. Surveys in combinatorics, 267:239-298, 1999. Google Scholar
  36. Elad Yehezkel. Selected neighbor degree tree realization. Master’s thesis, Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, 2020. Google Scholar
  37. K. A. Zaretskii. Constructing a tree on the basis of a set of distances between the hanging vertices. Uspekhi Mat. Nauk, 20:90-92, 1965. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact