## Abstract

The paper concerns the question of which properties of a graph are already determined by its degree sequence. The classic degree realization problem asks to characterize graphic sequences, i.e., sequences of positive integers, which are the degree sequence of some simple graph. Erdős and Gallai [11] solved this problem. Havel and Hakimi [14,17] provide a different characterization implying an algorithm to generate a realizing graph (if one exists). It is known that a graphic sequence can have several non-isomorphic realizations. We characterize graphic sequences where every realization has some given graph property P. Such sequences are called forcibly P-graphic. In particular, we present complete results characterizing forcibly (connected) bipartite, forcibly acyclic, and forcibly tree-graphic sequences. In those four models, we also characterize the sequences with a unique realizing graph, called unigraphic sequences. Finally, we address the problem of counting the number of sequences in each model.

Original language | English |
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Article number | 113460 |

Journal | Discrete Mathematics |

Volume | 346 |

Issue number | 7 |

DOIs | |

State | Published - Jul 2023 |

### Bibliographical note

Publisher Copyright:© 2023 Elsevier B.V.

### Funding

This work was supported by U.S-Israel Binational Science Foundation grant 2018043 .

Funders | Funder number |
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United States-Israel Binational Science Foundation | 2018043 |

## Keywords

- Acyclic graphs
- Bipartite graphs
- Degree sequence
- Forcibly P-graphic
- Graph realization
- Unique P-graphic