## 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 \(n\ge 7\), and the threshold on *n* is sharp. Using this result, we show that when \(n\ge 7\) 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 \(n\ge 6\), which are also sharp.

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## References

- 1.
Asciak, K.J., Francalanza, M.A., Lauri, J., Myrvold, W.: A survey of some open questions in reconstruction numbers. Ars Combinatoria

**97**, 443–456 (2010) - 2.
Bollobás, B.: Almost every graph has reconstruction number three. J. Graph Theory

**14**, 1–4 (1990) - 3.
Bondy, J.A.: A graph reconstructor’s manual. In: Surveys in Combinatorics (Guildford, 1991), London Mathematical Society Lecture Notes 166, pp. 221–252. Cambridge University Press, Cambridge (1991)

- 4.
Bondy, J.A., Hemminger, R.L.: Graph reconstruction—a survey. J. Graph Theory

**1**, 227–268 (1977) - 5.
Chernyak, Z.A.: Some additions to an article by B. Manvel: “Some basic observations on Kelly’s conjecture for graphs” (Russian). Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk.

**126**, 44–49 (1982) - 6.
Chinn, P.: A Graph with \(p\) Points and Enough Distinct \((p-2)\)-Order Subgraphs is Reconstructible, Recent Trends in Graph Theory Lecture Notes in Mathematics 186. Springer, New York (1971)

- 7.
Harary, F., Plantholt, M.: The graph reconstruction number. J. Graph Theory

**9**, 451–454 (1985) - 8.
Kelly, P.J.: On isometric transformations, PhD Thesis, University of Wisconsin-Madison (1942)

- 9.
Kelly, P.J.: A congruence theorem for trees. Pac. J. Math.

**7**, 961–968 (1957) - 10.
Lauri, J.: Pseudosimilarity in graphs—a survey. Ars Combinatoria

**46**, 77–95 (1997) - 11.
Lauri, J., Scapellato, R.: Topics in graph automorphism and reconstruction. In: London Mathematical Society Student Texts 54, Cambridge University Press, Cambridge (2003) (Second edition: London Math. Soc. Lect. Note Series 432, Cambridge Univ. Press, Cambridge, 2016)

- 12.
Maccari, A., Rueda, O., Viazzi, V.: A survey on edge reconstruction of graphs. J. Discrete Math. Sci. Cryptogr.

**5**, 1–11 (2002) - 13.
Manvel, B.: Some basic observations on Kelly’s conjecture for graphs. Discrete Math.

**8**, 181–185 (1974) - 14.
McMullen, B., Radziszowski, S.: Graph reconstruction numbers. J. Combinatorial Math. Combinatorial Comput.

**62**, 85–96 (2007) - 15.
Müller, V.: Probabilistic reconstruction from subgraphs. Comment. Math. Univ. Carolinae

**17**, 709–719 (1976) - 16.
Myrvold, W.J.: Ally and adversary reconstruction problems, Ph.D. thesis, Univ. Waterloo (1988)

- 17.
Nýdl, V.: Finite undirected graphs which are not reconstructible from their large cardinality subgraphs. Discrete Math.

**108**, 373–377 (1992) - 18.
Rivshin, D., Radziszowski, S.: Multi-vertex deletion graph reconstruction numbers. J. Combinatorial Math. Combinatorial Comput.

**78**, 303–321 (2011) - 19.
Spinoza, H., West, D.B.: Reconstruction from the deck of \(k\)-vertex induced subgraphs. J. Graph Theory

**90**, 497–522 (2019) - 20.
Taylor, R.: Reconstructing degree sequences from \(k\)-vertex-deleted subgraphs. Discrete Math.

**79**, 207–213 (1990) - 21.
Ulam, S.M.: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics 8. Interscience Publishers, Geneva (1960)

## Acknowledgements

A. V. Kostochka: Research supported in part by NSF grant DMS-1600592 and grants 18-01-00353A and 19-01-00682 of the Russian Foundation for Basic Research. D. B. West: Research supported by National Natural Science Foundation of China grants NNSFC 11871439 and 11971439. D. Zirlin: Research supported in part by Arnold O. Beckman Campus Research Board Award RB20003 of the University of Illinois at Urbana-Champaign

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Kostochka, A.V., Nahvi, M., West, D.B. *et al.* Degree Lists and Connectedness are 3-Reconstructible for Graphs with At Least Seven Vertices.
*Graphs and Combinatorics* (2020). https://doi.org/10.1007/s00373-020-02131-6

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### Keywords

- Graph reconstruction
- Deck
- Reconstructibility
- Connected

### Mathematics Subject Classification

- 05C60
- 05C07