Document Open Access Logo

Sparse Graphic Degree Sequences Have Planar Realizations

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

PDF
Thumbnail PDF

File

LIPIcs.MFCS.2024.18.pdf
  • Filesize: 0.91 MB
  • 17 pages

Document Identifiers

Author Details

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

Funding

Supported by a US-Israel BSF grant (2022205).

Cite As Get BibTex

Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. Sparse Graphic Degree Sequences Have Planar Realizations. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.18

Abstract

A sequence d = (d_1,d_2, …, d_n) of positive integers is graphic if it is the degree sequence of some simple graph G, and planaric if it is the degree sequence of some simple planar graph G. It is known that if ∑ d ≤ 2n - 2, then d has a realization by a forest, hence it is trivially planaric. In this paper, we seek bounds on ∑ d that guarantee that if d is graphic then it is also planaric. We show that this holds true when ∑ d ≤ 4n-4-2ω₁, where ω₁ is the number of 1’s in d. Conversely, we show that there are graphic sequences with ∑ d = 4n-2ω₁ that are non-planaric. For the case ω₁ = 0, we show that d is planaric when ∑ d ≤ 4n-4. Conversely, we show that there is a graphic sequence with ∑ d = 4n-2 that is non-planaric. In fact, when ∑ d ≤ 4n-6-2ω₁, d can be realized by a graph with a 2-page book embedding.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • Degree Sequences
  • Graph Algorithms
  • Graph Realization
  • Planar Graphs

Metrics

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

References

  1. Patrick Adams and Yuri Nikolayevsky. Planar bipartite biregular degree sequences. Discr. Math., 342:433-440, 2019. Google Scholar
  2. Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. Approximate realizations for outerplanaric degree sequences. In Proc. 35th IWOCA, 2024. Google Scholar
  3. Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. On key parameters affecting the realizability of degree sequences. In Proc. 49th MFCS, 2024. Google Scholar
  4. Frank Bernhart and Paul C Kainen. The book thickness of a graph. Journal of Combinatorial Theory, Series B, 27(3):320-331, 1979. Google Scholar
  5. Prosenjit Bose, Vida Dujmović, Danny Krizanc, Stefan Langerman, Pat Morin, David R Wood, and Stefanie Wuhrer. A characterization of the degree sequences of 2-trees. Journal of Graph Theory, 58(3):191-209, 2008. Google Scholar
  6. SA Choudum. Characterization of forcibly outerplanar graphic sequences. In Combinatorics and Graph Theory, pages 203-211. Springer, 1981. Google Scholar
  7. Paul Erdös and Tibor Gallai. Graphs with prescribed degrees of vertices [hungarian]. Matematikai Lapok, 11:264-274, 1960. Google Scholar
  8. Stefano Fanelli. On a conjecture on maximal planar sequences. Journal of Graph Theory, 4(4):371-375, 1980. Google Scholar
  9. Stefano Fanelli. An unresolved conjecture on nonmaximal planar graphical sequences. Discrete Mathematics, 36(1):109-112, 1981. Google Scholar
  10. Gautam Gupta, Puneet Joshi, and Amitabha Tripathi. Graphic sequences of trees and a problem of Frobenius. Czechoslovak Math. J., 57:49-52, 2007. Google Scholar
  11. 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
  12. V. Havel. A remark on the existence of finite graphs [in Czech]. Casopis Pest. Mat., 80:477-480, 1955. Google Scholar
  13. AF Hawkins, AC Hill, JE Reeve, and JA Tyrrell. On certain polyhedra. The Mathematical Gazette, 50(372):140-144, 1966. Google Scholar
  14. Kyle F. Jao and Douglas B. West. Vertex degrees in outerplanar graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 82:229-239, 2012. Google Scholar
  15. Zepeng Li and Yang Zuo. On the degree sequences of maximal out erplanar graphs. Ars Combinatoria, 2018. Google Scholar
  16. A Ramachandra Rao. Degree sequences of cacti. In Combinatorics and Graph Theory: Proceedings of the Symposium Held at the Indian Statistical Institute, Calcutta, February 25-29, 1980, pages 410-416. Springer, 1981. Google Scholar
  17. 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
  18. E. F. Schmeichel and S. L. Hakimi. On planar graphical degree sequences. SIAM J. Applied Math., 32:598-609, 1977. Google Scholar
  19. 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
  20. D.B. West. Introduction to graph theory. Prentice Hall, 2001. Google Scholar
  21. Mihalis Yannakakis. Four pages are necessary and sufficient for planar graphs. In 18th annual ACM symposium on theory of computing, pages 104-108, 1986. 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