Skip to main content
Springer Nature Link
Log in

Ordered Sets that are Reconstructible from Two Cards and the Number of Comparabilities

  • Published:
Order Aims and scope Submit manuscript

Abstract

This paper proves that, if there is an ordered set P that is not reconstructible, then each of its two-point-deleted subsets must be isomorphic to another one, or, it must violate a condition that is related to, but weaker than, rigidity. The conditions are inspired by an argument by Bollobas and they provide a connection between positive reconstruction results and partial counterexamples that was, so far, nonexistent in order reconstruction.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berge, C., Rado, R.: Note on isomorphic hypergraphs and some extensions of Whitney’s theorem to families of sets. J. Comb. Theory Ser. B 13, 226–241 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bollobás, B.: Almost every graph has reconstruction number three. J. Graph Theory 14, 1–4 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bondy, J.A.: A graph reconstructor’s manual. Surveys in Combinatorics. In: Proceedings of the 13th British Combinatorial Conference, pp. 221–252 (1991)

  4. Bondy, J.A., Hemminger, R.L.: Graph reconstruction—a survey. J. Graph Theory 1, 227–268 (1976)

    Article  MathSciNet  Google Scholar 

  5. Godsil, C.D., Kocay, W.L.: Constructing graphs with pairs of pseudosimilar vertices. J. Comb. Theory Ser. B 32, 146–155 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kratsch, D., Rampon, J.-X.: A counterexample about poset reconstruction. Order 11, 95–96 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kratsch, D., Rampon, J.-X.: Towards the reconstruction of posets. Order 11, 317–341 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lauri, J.: Graph reconstruction–some techniques and new problems (Proceedings of the 1st Catania International Combinatorial Conference on Graphs, Steiner Systems, and their Applications, Catania, 1986). Ars Combin. 24, 35–61 (1987)

    MathSciNet  Google Scholar 

  9. Manvel, B.: Reconstruction of graphs: progress and prospects (250th Anniversary Conference on Graph Theory, Fort Wayne, IN, 1986). Congr. Numer. 63, 177–187 (1988)

    MathSciNet  Google Scholar 

  10. Rampon, J.-X.: What is reconstruction for ordered sets? Discret. Math. 291, 191–233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sands, B.: Unsolved problems. Order 1, 311–313 (1985)

    Article  MathSciNet  Google Scholar 

  12. Schröder, B.: Examples on ordered set reconstruction. Order 19, 283–294 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Schröder, B.: Ordered Sets—An Introduction. Birkhäuser. Boston, Basel, Berlin (2003)

    Book  MATH  Google Scholar 

  14. Schröder, B.: On ordered sets with isomorphic marked maximal cards. Order 20, 299–327 (2003)

    Article  MathSciNet  Google Scholar 

  15. Schröder, B.: More examples on ordered set reconstruction. Discret. Math. 280, 149–163 (2004)

    Article  MATH  Google Scholar 

  16. Schröder, B.: Pseudo-similar points in ordered sets. Discret. Math. 310, 2815–2823 (2010)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Program of Mathematics and Statistics, Louisiana Tech University, P.O. Box 10348, Ruston, LA, 71272, USA

    Bernd S. W. Schröder

Authors
  1. Bernd S. W. Schröder
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Bernd S. W. Schröder.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schröder, B.S.W. Ordered Sets that are Reconstructible from Two Cards and the Number of Comparabilities. Order 31, 365–371 (2014). https://doi.org/10.1007/s11083-013-9306-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-013-9306-4

Keywords

Mathematics Subject Classifications (2010)