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# On Key Parameters Affecting the Realizability of Degree Sequences (Invited Paper)

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## Cite As

Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. On Key Parameters Affecting the Realizability of Degree Sequences (Invited Paper). In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.MFCS.2024.1

## Abstract

Call a sequence d = (d_1,d_2, …, d_n) of positive integers graphic, planaric, outer-planaric, or forestic if it is the degree sequence of some arbitrary, planar, outer-planar, or cycle-free graph G, respectively. The two extreme classes of graphic and forestic sequences were given full characterizations. (The latter has a particularly simple criterion: d is forestic if and only if its volume, ∑ d ≡ ∑_i d_i, satisfies ∑ d ≤ 2n - 2.) In contrast, the problems of fully characterizing planaric and outer-planaric degree sequences are still open. In this paper, we discuss the parameters affecting the realizability of degree sequences by restricted classes of sparse graph, including planar graphs, outerplanar graphs, and some of their subclasses (e.g., 2-trees and cactus graphs). A key parameter is the volume of the sequence d, namely, ∑ d which is twice the number of edges in the realizing graph. For planar graphs, for example, an obvious consequence of Euler’s theorem is that an n-element sequence d satisfying ∑ d > 4n-6 cannot be planaric. Hence, ∑ d ≤ 4n-6 is a necessary condition for d to be planaric. What about the opposite direction? Is there an upper bound on ∑ d that guarantees that if d is graphic then it is also planaric. Does the answer depend on additional parameters? The same questions apply also to sub-classes of the planar graphs. A concrete example that is illustrated in the technical part of the paper is the class of outer-planaric degree sequences. Denoting the number of 1’s in d by ω₁, we show that for a graphic sequence d, if ω₁ = 0 then d is outer-planaric when ∑ d ≤ 3n-3, and if ω₁ > 0 then d is outer-planaric when ∑ d ≤ 3n-ω₁-2. Conversely, we show that there are graphic sequences that are not outer-planaric with ω₁ = 0 and ∑ d = 3n-2, as well as ones with ω₁ > 0 and ∑ d = 3n-ω₁-1.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
##### Keywords
• Degree Sequences
• Graph Algorithms
• Graph Realization
• Outer-planar Graphs

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## References

1. Patrick Adams and Yuri Nikolayevsky. Planar bipartite biregular degree sequences. Discr. Math., 342:433-440, 2019.
2. Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. Degree realization by bipartite cactus graphs. Unpublished Manuscript.
3. Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. Approximate realizations for outerplanaric degree sequences. In Proc. 35th IWOCA, 2024.
4. Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz. Sparse graphic degree sequences have planar realizations. In Proc. 49th MFCS, 2024.
5. Lowell W Beineke and Edward F Schmeichel. Degrees and cycles in graphs. Annals of the New York Academy of Sciences, 319(1):64-70, 1979.
6. F. T. Boesch and F. Harary. Unicyclic realizability of a degree list. Networks, 8:93-96, 1978.
7. 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. JGT, 58:191-209, 2008.
8. SA Choudum. Characterization of forcibly outerplanar graphic sequences. In Combinatorics and Graph Theory, pages 203-211. Springer, 1981.
9. Paul Erdös and Tibor Gallai. Graphs with prescribed degrees of vertices [hungarian]. Matematikai Lapok, 11:264-274, 1960.
10. Stefano Fanelli. On a conjecture on maximal planar sequences. Journal of Graph Theory, 4(4):371-375, 1980.
11. Stefano Fanelli. An unresolved conjecture on nonmaximal planar graphical sequences. Discrete Mathematics, 36(1):109-112, 1981.
12. Gautam Gupta, Puneet Joshi, and Amitabha Tripathi. Graphic sequences of trees and a problem of Frobenius. Czechoslovak Math. J., 57:49-52, 2007.
13. 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.
14. V. Havel. A remark on the existence of finite graphs [in Czech]. Casopis Pest. Mat., 80:477-480, 1955.
15. AF Hawkins, AC Hill, JE Reeve, and JA Tyrrell. On certain polyhedra. The Mathematical Gazette, 50(372):140-144, 1966.
16. Kyle F. Jao and Douglas B. West. Vertex degrees in outerplanar graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 82:229-239, 2012.
17. Zepeng Li and Yang Zuo. On the degree sequences of maximal outerplanar graphs. Ars Comb., 140:237-250, 2018.
18. Zvi Lotker, Debapriyo Majumdar, N. S. Narayanaswamy, and Ingmar Weber. Sequences characterizing k-trees. In 12th COCOON, volume 4112 of LNCS, pages 216-225, 2006.
19. Md Tahmidur Rafid, Rabeeb Ibrat, and Md Saidur Rahman. Generating scale-free outerplanar networks. In Int. Computer Symp., pages 156-166. Springer, 2022.
20. A. Ramachandra Rao. Degree sequences of cacti. In Combinatorics and Graph Theory, LNM, pages 410-416, 1981.
21. 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.
22. S. Rengarajan and C. E. Veni Madhavan. Stack and queue number of 2-trees. In 1st COCOON, volume 959 of LNCS, pages 203-212, 1995.
23. E. F. Schmeichel and S. L. Hakimi. On planar graphical degree sequences. SIAM J. Applied Math., 32:598-609, 1977.
24. Maciej M Sysło. Characterizations of outerplanar graphs. Discrete Mathematics, 26(1):47-53, 1979.
25. D.B. West. Introduction to graph theory. Prentice Hall, 2001.