### Abstract

The k-deck of a graph is the multiset of its subgraphs induced by k vertices. A graph or graph property is l-reconstructible if it is determined by the deck of subgraphs obtained by deleting l vertices. We show that the degree list of an n-vertex graph is 3-reconstructible when n≥ 7 , and the threshold on n is sharp. Using this result, we show that when n≥ 7 the (n- 3) -deck also determines whether an n-vertex graph is connected; this is also sharp. These results extend the results of Chernyak and Manvel, respectively, that the degree list and connectedness are 2-reconstructible when n≥ 6 , which are also sharp.

Original language | English (US) |
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Pages (from-to) | 491-501 |

Number of pages | 11 |

Journal | Graphs and Combinatorics |

Volume | 36 |

Issue number | 3 |

DOIs | |

State | Published - May 2020 |

### Keywords

- Connected
- Deck
- Graph reconstruction
- Reconstructibility

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Kostochka, A. V., Nahvi, M., West, D. B., & Zirlin, D. (2020). Degree Lists and Connectedness are 3-Reconstructible for Graphs with At Least Seven Vertices.

*Graphs and Combinatorics*,*36*(3), 491-501. https://doi.org/10.1007/s00373-020-02131-6