eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-23
18:1
18:17
10.4230/LIPIcs.MFCS.2024.18
article
Sparse Graphic Degree Sequences Have Planar Realizations
Bar-Noy, Amotz
1
Böhnlein, Toni
2
Peleg, David
3
Ran, Yingli
3
https://orcid.org/0009-0008-5819-7543
Rawitz, Dror
4
City University of New York (CUNY), NY, USA
Huawei, Zurich, Switzerland
Weizmann Institute of Science, Rehovot, Israel
Bar Ilan University, Ramat-Gan, Israel
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol306-mfcs2024/LIPIcs.MFCS.2024.18/LIPIcs.MFCS.2024.18.pdf
Degree Sequences
Graph Algorithms
Graph Realization
Planar Graphs