## 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} di, 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 outerplanaric degree sequences. Denoting the number of 1’s in d by ω1, we show that for a graphic sequence d, if ω_{1} = 0 then d is outer-planaric when ∑ d ≤ 3n − 3, and if ω_{1} > 0 then d is outer-planaric when ∑ d ≤ 3n− ω_{1} − 2. Conversely, we show that there are graphic sequences that are not outer-planaric with ω_{1} = 0 and ∑ d = 3n − 2, as well as ones with ω_{1} > 0 and ∑ d = 3n − ω_{1} − 1.

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
- Outer-planar Graphs