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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
Bollobás, B.: Almost every graph has reconstruction number three. J. Graph Theory 14, 1–4 (1990)
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)
Bondy, J.A., Hemminger, R.L.: Graph reconstruction—a survey. J. Graph Theory 1, 227–268 (1977)
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)
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)
Harary, F., Plantholt, M.: The graph reconstruction number. J. Graph Theory 9, 451–454 (1985)
Kelly, P.J.: On isometric transformations, PhD Thesis, University of Wisconsin-Madison (1942)
Kelly, P.J.: A congruence theorem for trees. Pac. J. Math. 7, 961–968 (1957)
Lauri, J.: Pseudosimilarity in graphs—a survey. Ars Combinatoria 46, 77–95 (1997)
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)
Maccari, A., Rueda, O., Viazzi, V.: A survey on edge reconstruction of graphs. J. Discrete Math. Sci. Cryptogr. 5, 1–11 (2002)
Manvel, B.: Some basic observations on Kelly’s conjecture for graphs. Discrete Math. 8, 181–185 (1974)
McMullen, B., Radziszowski, S.: Graph reconstruction numbers. J. Combinatorial Math. Combinatorial Comput. 62, 85–96 (2007)
Müller, V.: Probabilistic reconstruction from subgraphs. Comment. Math. Univ. Carolinae 17, 709–719 (1976)
Myrvold, W.J.: Ally and adversary reconstruction problems, Ph.D. thesis, Univ. Waterloo (1988)
Nýdl, V.: Finite undirected graphs which are not reconstructible from their large cardinality subgraphs. Discrete Math. 108, 373–377 (1992)
Rivshin, D., Radziszowski, S.: Multi-vertex deletion graph reconstruction numbers. J. Combinatorial Math. Combinatorial Comput. 78, 303–321 (2011)
Spinoza, H., West, D.B.: Reconstruction from the deck of \(k\)-vertex induced subgraphs. J. Graph Theory 90, 497–522 (2019)
Taylor, R.: Reconstructing degree sequences from \(k\)-vertex-deleted subgraphs. Discrete Math. 79, 207–213 (1990)
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
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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 36, 491–501 (2020). https://doi.org/10.1007/s00373-020-02131-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-020-02131-6