## 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 − 2ω_{1}, where ω_{1} is the number of 1’s in d. Conversely, we show that there are graphic sequences with ∑ d = 4n − 2ω_{1} that are non-planaric. For the case ω_{1} = 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 − 2ω_{1}, d can be realized by a graph with a 2-page book embedding.

Original language | English |
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Title of host publication | 49th International Symposium on Mathematical Foundations of Computer Science, MFCS 2024 |

Editors | Rastislav Kralovic, Antonin Kucera |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959773355 |

DOIs | |

State | Published - Aug 2024 |

Event | 49th International Symposium on Mathematical Foundations of Computer Science, MFCS 2024 - Bratislava, Slovakia Duration: 26 Aug 2024 → 30 Aug 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 306 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 49th International Symposium on Mathematical Foundations of Computer Science, MFCS 2024 |
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Country/Territory | Slovakia |

City | Bratislava |

Period | 26/08/24 → 30/08/24 |

### Bibliographical note

Publisher Copyright:© Amotz Bar-Noy, Toni Böhnlein, David Peleg, Yingli Ran, and Dror Rawitz.

## Keywords

- Degree Sequences
- Graph Algorithms
- Graph Realization
- Planar Graphs