Skip to main content
SpringerLink
Log in

Dimension of Restricted Classes of Interval Orders

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Rabinovitch showed in 1978 that the interval orders having a representation consisting of only closed unit intervals have order dimension at most \(3\). This article shows that the same dimension bound applies to two other classes of posets: those having a representation consisting of unit intervals (but with a mixture of open and closed intervals allowed) and those having a representation consisting of closed intervals with lengths in \(\left\{ 0,1 \right\}\).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Availability of Data and Material

Not applicable.

Code Availability

Not applicable.

References

  1. Bogart, K.P., Rabinovich, I., Trotter, W.T., Jr.: A bound on the dimension of interval orders. J. Combin. Theory Ser. A 21(3), 319–328 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bosek, B., Kloch, K., Krawczyk, T., Micek, P.: On-line version of Rabinovitch theorem for proper intervals. Discrete Math. 312(23), 3426–3436 (2012). https://doi.org/10.1016/j.disc.2012.02.008

    Article  MathSciNet  MATH  Google Scholar 

  3. Boyadzhiyska, S., Isaak, G., Trenk, A.N.: A simple proof characterizing interval orders with interval lengths between 1 and k. Involve J. Math. 11(5), 893–900 (2018). https://doi.org/10.2140/involve.2018.11.893

    Article  MathSciNet  MATH  Google Scholar 

  4. Boyadzhiyska, S., Isaak, G., Trenk, A.N.: Interval orders with two interval lengths. Discrete Appl. Math. 267, 52–63 (2019). https://doi.org/10.1016/j.dam.2019.06.021

    Article  MathSciNet  MATH  Google Scholar 

  5. Cerioli, M.R., Oliveira, F.D.S., Szwarcfiter, J.L.: On counting interval lengths of interval graphs. Discrete Appl. Math. 159(7), 532–543 (2011). https://doi.org/10.1016/j.dam.2010.07.006

    Article  MathSciNet  MATH  Google Scholar 

  6. Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. J. Math. Psychol. 7, 144–149 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fishburn, P.C., Graham, R.L.: Classes of interval graphs under expanding length restrictions. J. Graph Theory 9(4), 459–472 (1985). https://doi.org/10.1002/jgt.3190090405

    Article  MathSciNet  MATH  Google Scholar 

  8. Füredi, Z., Hajnal, P., Rödl, V., Trotter, W.T.: Interval orders and shift graphs. In: Hajnal, A., Sos, V.T. (eds.) Sets, Graphs and Numbers, Colloquium Mathematical Society János Bolyai, vol. 60, pp. 297–313 (1991)

  9. Golumbic, M.C., Trenk, A.N.: Tolerance Graphs, Cambridge Studies in Advanced Mathematics, vol. 89. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  10. Hoşten, S., Morris, W.D.: The order dimension of the complete graph. Discrete Math. 201(1), 133–139 (1999). https://doi.org/10.1016/S0012-365X(98)00315-X

    Article  MathSciNet  MATH  Google Scholar 

  11. Isaak, G.: Bounded discrete representations of interval orders. Discrete Appl. Math. 44(1), 157–183 (1993). https://doi.org/10.1016/0166-218X(93)90229-H

    Article  MathSciNet  MATH  Google Scholar 

  12. Joos, F.: A characterization of mixed unit interval graphs. J. Graph Theory 79(4), 267–281 (2015). https://doi.org/10.1002/jgt.21831

    Article  MathSciNet  MATH  Google Scholar 

  13. Kelly, D.: The 3-irreducible partially ordered sets. Can. J. Math. 29(2), 367–383 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kleitman, D., Markowsky, G.: On Dedekind’s problem: the number of isotone Boolean functions. II. Trans. Am. Math. Soc. 213, 373–390 (1975). https://doi.org/10.2307/1998052

    Article  MathSciNet  MATH  Google Scholar 

  15. Köbler, J., Kuhnert, S., Watanabe, O.: Interval graph representation with given interval and intersection lengths. J. Discrete Algorithms 34, 108–117 (2015). https://doi.org/10.1016/j.jda.2015.05.011

    Article  MathSciNet  MATH  Google Scholar 

  16. Pe’er, I., Shamir, R.: Interval graphs with side (and size) constraints. In: P. Spirakis (ed.) Algorithms—ESA ’95, Lecture Notes in Computer Science, pp. 142–154. Springer, Berlin, Heidelberg (1995). https://doi.org/10.1007/3-540-60313-1_140

  17. Pe’er, I., Shamir, R.: Realizing interval graphs with size and distance constraints. SIAM J. Discrete Math. 10(4), 662–687 (1997). https://doi.org/10.1137/S0895480196306373

    Article  MathSciNet  MATH  Google Scholar 

  18. Rabinovitch, I.: The dimension-theory of semiorders and interval-orders. Ph.D. thesis, Dartmouth College (1973)

  19. Rabinovitch, I.: The dimension of semiorders. J. Combin. Theory Ser. A 25(1), 50–61 (1978). https://doi.org/10.1016/0097-3165(78)90030-4

    Article  MathSciNet  MATH  Google Scholar 

  20. Rautenbach, D., Szwarcfiter, J.L.: Unit interval graphs of open and closed intervals. J. Graph Theory 72(4), 418–429 (2013). https://doi.org/10.1002/jgt.21650

    Article  MathSciNet  MATH  Google Scholar 

  21. Scott, D., Suppes, P.: Foundational aspects of theories of measurement. J. Symb. Logic 23, 113–128 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  22. Shuchat, A., Shull, R., Trenk, A.N.: Unit interval orders of open and closed intervals. Order 33(1), 85–99 (2016). https://doi.org/10.1007/s11083-015-9354-z

    Article  MathSciNet  MATH  Google Scholar 

  23. Shuchat, A., Shull, R., Trenk, A.N., West, L.C.: Unit mixed interval graphs. Congr. Numer. 221, 189–223 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Trotter, W.T.: Combinatorics and Partially Ordered Sets: Dimension Theory. Johns Hopkins Series in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1992)

    MATH  Google Scholar 

  25. Trotter, W.T.: New perspectives on interval orders and interval graphs. In: Surveys in Combinatorics, 1997 (London), London Mathematical Society. Lecture Note Series, vol. 241, pp. 237–286. Cambridge University Press, Cambridge (1997)

  26. Trotter, W.T.: Personal communication (2020)

  27. Trotter, W.T., Jr., Moore, J.I., Jr.: Characterization problems for graphs, partially ordered sets, lattices, and families of sets. Discrete Math. 16(4), 361–381 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  28. Trotter, W.T., Jr., Moore, J.I., Jr.: The dimension of planar posets. J. Combin. Theory Ser. B 22(1), 54–67 (1977)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank an anonymous referee for valuable suggestions that improved the paper.

Funding

This work was supported by a grant from the Simons Foundation (#426725, Ann Trenk).

Author information

Authors and Affiliations

  1. University of Wisconsin-Madison, Madison, WI, USA

    Mitchel T. Keller

  2. Department of Mathematics, Wellesley College, Wellesley, MA, USA

    Ann N. Trenk

  3. Pacific Northwest National Laboratory, Richland, WA, USA

    Stephen J. Young

Authors
  1. Mitchel T. Keller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ann N. Trenk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephen J. Young
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mitchel T. Keller.

Ethics declarations

Conflict of interest

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

PNNL Information Release: PNNL-SA-152654.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Keller, M.T., Trenk, A.N. & Young, S.J. Dimension of Restricted Classes of Interval Orders. Graphs and Combinatorics 38, 137 (2022). https://doi.org/10.1007/s00373-022-02543-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00373-022-02543-6

Keywords

Mathematics Subject Classification

Search

Navigation