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Degree Lists and Connectedness are 3-Reconstructible for Graphs with At Least Seven Vertices

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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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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Correspondence to Douglas B. West.

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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