Skip to main content

Volume 18 Supplement 2

GraphCliques

The interval count of interval graphs and orders: a short survey

Abstract

The interval count problem determines the smallest number of interval lengths needed in order to represent an interval model of a given interval graph or interval order. Despite the large number of studies about interval graphs and interval orders, surprisingly only a few results on the interval count problem are known. In this work, we provide a short survey about the interval count and related problems. a graph and the number of its maximal cliques.

References

  1. 1.

    Allen JF (1990) Maintaining knowledge about temporal intervals. Morgan Kaufmann, San Mateo

    Book  Google Scholar 

  2. 2.

    Allen JF, Kautz HA, Pelavin RN, Tenenberg JD (1991) Reasoning about plans. Morgan Kaufmann, San Mateo

    MATH  Google Scholar 

  3. 3.

    Benzer S (1959) On the topology of the genetic fine structure. Proc Natl Acad Sci USA 45(11):1607–1620

    Article  Google Scholar 

  4. 4.

    Bogart KP, West DB (1999) A short proof that “proper = unit”. Discrete Math 201(1–3):21–23

    MATH  MathSciNet  Article  Google Scholar 

  5. 5.

    Bondy JA, Murty USR (2008) Graph theory. Springer, Berlin

    MATH  Book  Google Scholar 

  6. 6.

    Brandstädt A, Le VB, Szymczak T, Siegemund F Information system on graph class inclusions. http://wwwteo.informatik.uni-rostock.de/isgci/

  7. 7.

    Carrano AV (1988) Establishing the order to human chromosome-specific DNA fragments. In: Woodhead A, Barnhart B (eds) Biotechnology and the human genome. Plenum Press, New York, pp 37–50

    Chapter  Google Scholar 

  8. 8.

    Cerioli MR, Szwarcfiter JL (2006) Characterizing intersection graphs of substars of a star. Ars Comb 79:21–31

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Cerioli MR, Oliveira FS, Szwarcfiter JL (2011) On counting interval lengths of interval graphs. Discrete Appl Math 159(7):532–543

    MATH  MathSciNet  Article  Google Scholar 

  10. 10.

    Coombs CH, Smith JEK (1973) On the detection of structures in attitudes and developmental processes. Psychol Rev 80:337–351

    Article  Google Scholar 

  11. 11.

    Corneil DG (2004) A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs. Discrete Appl Math 138(3):371–379

    MATH  MathSciNet  Article  Google Scholar 

  12. 12.

    Corneil DG, Kim H, Natarajan S, Olariu S, Sprague AP (1995) Simple linear time recognition of unit interval graphs. Inf Process Lett 55(2):99–104

    MATH  MathSciNet  Article  Google Scholar 

  13. 13.

    de Figueiredo CMH, Meidanis J, de Mello CP (1995) A linear-time algorithm for proper interval graph recognition. Inf Process Lett 56(3):179–184

    MATH  Article  Google Scholar 

  14. 14.

    Deng X, Hell P, Huang J (1996) Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs. SIAM J Comput 25(2):390–403

    MATH  MathSciNet  Article  Google Scholar 

  15. 15.

    Fishburn PC (1985) Interval orders and interval graphs. Wiley, New York

    MATH  Google Scholar 

  16. 16.

    Gardi F (2007) The Roberts characterization of proper and unit interval graphs. Discrete Math 307(22):2906–2908

    MATH  MathSciNet  Article  Google Scholar 

  17. 17.

    Golumbic MC (2004) Algorithmic graph theory and perfect graphs, 2nd edn. Amsterdam, Elsevier

    MATH  Google Scholar 

  18. 18.

    Greenough T (1974) The representation and enumeration of interval orders. PhD thesis, Darthmouth College

  19. 19.

    Hell P, Huang J (2005) Certifying lexbfs recognition algorithms for proper interval graphs and proper interval bigraphs. SIAM J Discrete Math 18(3):554–570

    MATH  MathSciNet  Article  Google Scholar 

  20. 20.

    Isaak G (1993) Discrete interval graphs with bounded representation. Discrete Appl Math 33:157–183

    MathSciNet  Article  Google Scholar 

  21. 21.

    Karp RM (1993) Mapping the genome: some combinatorial problems arising in molecular biology. In: STOC ’93. ACM, New York, pp 278–285

    Chapter  Google Scholar 

  22. 22.

    Kendall DG (1969) Incidence matrices, interval graphs, and seriation in archaeology. Pac J Math 28:565–570

    MATH  MathSciNet  Article  Google Scholar 

  23. 23.

    Leibowitz R (1978) Interval counts and threshold graphs. PhD thesis, Rutgers University

  24. 24.

    Leibowitz R, Assmann SF, Peck GW (1982) The interval count of a graph. SIAM J Algebr Discrete Methods 3(4):485–494

    MATH  MathSciNet  Article  Google Scholar 

  25. 25.

    Nökel K (1991) Temporally distributed symptoms in technical diagnosis. Springer, Berlin

    MATH  Book  Google Scholar 

  26. 26.

    Papadimitriou CH, Yannakakis M (1979) Scheduling interval-ordered tasks. SIAM J Comput 8(3):405–409

    MATH  MathSciNet  Article  Google Scholar 

  27. 27.

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

    MATH  MathSciNet  Article  Google Scholar 

  28. 28.

    Roberts FS (1969) Indifference graphs. In: Harary F (ed) Proof techniques in graph theory. Academic Press, San Diego, pp 139–146

    Google Scholar 

  29. 29.

    Roberts FS (1976) Discrete mathematical models with applications to social, biological, and environmental problems. Prentice Hall, New York

    MATH  Google Scholar 

  30. 30.

    Skrien D (1984) Chronological orderings of interval graphs. Discrete Appl Math 8:69–83

    MATH  MathSciNet  Article  Google Scholar 

  31. 31.

    Trotter WT (1988) Interval graphs, interval orders, and their generalizations. In: Applications of discrete mathematics. SIAM, Philadelphia, pp 45–58

    Google Scholar 

  32. 32.

    Trotter WT (1992) Combinatorics and partially ordered sets. The Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  33. 33.

    Ward SA, Halstead RH (1990) Computation structures. MIT Press & McGraw-Hill, Cambridge

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Fabiano de S. Oliveira.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Cerioli, M.R., de S. Oliveira, F. & Szwarcfiter, J.L. The interval count of interval graphs and orders: a short survey. J Braz Comput Soc 18, 103–112 (2012). https://doi.org/10.1007/s13173-011-0047-1

Download citation

Keywords

  • Interval count
  • Interval lengths
  • Number of lengths