, Volume 20, Issue 2, pp 151–171 | Cite as

Extendability of Cyclic Orders

  • Samuel Fiorini
  • Peter C. Fishburn


A cyclic order is a ternary relation that satisfies ternary transitivity and asymmetry conditions. Such a ternary relation is extendable if it is included in a complete cyclic order on the same ground set. Unlike the case of linear extensions of partial orders, a cyclic order need not be extendable. The extension problem for cyclic orders is to determine if a cyclic order is extendable. This problem is known to be NP-complete. We introduce a class of cyclic orders in which the extension problem can be solved in polynomial time. The class provides many explicit examples of nonextendable cyclic orders that were not previously known, including a nonextendable cyclic order on seven points. Let μ be the maximum cardinality of a ground set on which all cyclic orders are extendable. It has been shown that μ≤9. We prove that μ=6. This answers a question of Novák. In addition, we characterize the nonextendable cyclic orders on seven and eight points. Our results are intimately related to irreducible partially ordered set of order dimension three, and to fractional vertices of generalized transitive tournament polytopes. As by-products, we obtain a characterization of cyclically ordered sets of dimension two, and a new proof of a theorem of Dridi on small linear ordering polytopes.

cyclic orders poset dimension polytopes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alles, P.: Erweiterungen, Diagramme und Dimension zyklischer Ordnungen, Ph.D. Thesis, Technische Hochschule, Darmstadt, Germany, 1986.zbMATHGoogle Scholar
  2. 2.
    Alles, P., Nešetřil, J. and Poljak, S.: Extendability, dimensions, and diagrams of cyclic orders, SIAM J. Discrete Math. 4(4) (1991), 453–471.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Borobia, A.: (0, 12, 1) matrices which are extreme points of the generalized transitive tournament polytope, Linear Algebra Appl. 220 (1995), 97–110.zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Borobia, A. and Chumillas, V.: *-graphs of vertices of the generalized transitive tournament polytope, Discrete Math. 179(1#x2013;3) (1998), 49–57.zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Brualdi, R. and Hwang, S.-G.: Generalized transitive tournaments and doubly stochastic matrices, Linear Algebra Appl. 172 (1992), 151–168.zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Chajda, I. and Novák, V.: On extensions of cyclic orders, Československá Akademie Věd. Časopis pro Pěstování Matematiky 110 (1985), 116–121.zbMATHGoogle Scholar
  7. 7.
    Cruse, A.: On removing a vertex from the assignment polytope, Linear Algebra Appl. 26 (1979), 45–57.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dridi, T.: Sur les distributions binaires associées à des distributions ordinales, Mathématiques et Sciences Humaines 69 (1980), 15–31.zbMATHMathSciNetGoogle Scholar
  9. 9.
    Dushnik, B. and Miller, E.: Partially ordered sets, Amer. J. Math. 63 (1941), 600–610.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fiorini, S.: Polyhedral combinatorics of order polytopes, Ph.D. Thesis, Université Libre de Bruxelles, Brussels, Belgium, 2001.Google Scholar
  11. 11.
    Fishburn, P.: Decomposing weighted digraphs into sum of chains, Discrete Appl. Math. 16 (1987), 223–238.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Fishburn, P.: Binary probabilities induced by rankings, SIAM J. Discrete Math. 3 (1990), 478–488.zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Fishburn, P.: Induced binary probabilities and the linear ordering polytope: A status report, Math. Social Sci. 23 (1992), 67–80.zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Fishburn, P. and Woodall, D.: Cycle orders, Order 16 (1999), 149–164.zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Galil, Z. and Megiddo, N.: Cyclic ordering is NP-complete, Theoret. Comput. Sci. 5 (1977), 179–182.MathSciNetCrossRefGoogle Scholar
  16. 16.
    Ganter, B.: Two basic algorithms in concept analysis, Technical Report 831, Technische Hochschule Darmstadt, 1984.Google Scholar
  17. 17.
    Ganter, B. and Reuter, K.: Finding all closed sets: A general approach, Order 8 (1991), 283–290.zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Goemans, M. and Hall, L.: The strongest facets of the acyclic subgraph polytope are unknown, Integer Programming and Optimization 1084 (1996), 415–429.MathSciNetGoogle Scholar
  19. 19.
    Heyting, A.: Axiomatic Projective Geometry, 2nd edn, Bibliotheca Mathematica, North-Holland, Amsterdam, 1980.zbMATHGoogle Scholar
  20. 20.
    Hiraguchi, T.: On the dimension of orders, Science Rep. Kanazawa Univ. 1 (1955).Google Scholar
  21. 21.
    Huntington, E.: A set of independent postulates for cyclic order, Proc. Nat. Acad. Sci. U.S.A. 2 (1916), 630–631.zbMATHCrossRefGoogle Scholar
  22. 22.
    Huntington, E.: Set of completely independent postulates for cyclic order, Proc. Nat. Acad. Sci. U.S.A. 10 (1924), 74–78.zbMATHCrossRefGoogle Scholar
  23. 23.
    Kelly, D.: The 3-irreducible partially ordered sets, Canad. J. Math. 29(2) (1977), 367–383.zbMATHMathSciNetGoogle Scholar
  24. 24.
    Megiddo, N.: Partial and complete cyclic orders, Bull. Amer. Math. Soc. 82 (1976), 274–276.zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Megiddo, N.: Mixtures of order matrices and generalized order matrices, Discrete Math. 19 (1977), 177–181.zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Mitchell, J. and Borchers, B.: Solving linear ordering problems with a combined interior point/simplex cutting plane algorithm, In: H. F. et al. (eds), High Performance Optimization, 2000, pp. 349–366.Google Scholar
  27. 27.
    Nalivaiko, V.: Das lineare Ordnungsproblem, Ph.D. Thesis, Falkultät für Mathematik, Universität Magdeburg, Magdeburg, Germany, 1999.Google Scholar
  28. 28.
    Novák, V.: Cyclically ordered sets, Czechoslovak Math. J. 32 (1982), 460–473.zbMATHMathSciNetGoogle Scholar
  29. 29.
    Novák, V.: On some minimal problem, Arch. Math. (Brno) 10 (1984), 95–100.Google Scholar
  30. 30.
    Nutov, Z. and Penn, M.: On non-{0, 12, 1} extreme points of the generalized transitive tournament polytope, Linear Algebra Appl. 233 (1996), 149–159.zbMATHMathSciNetCrossRefGoogle Scholar
  31. 31.
    Pickert, G.: Projektive Ebenen, Springer, Berlin, 1955.zbMATHGoogle Scholar
  32. 32.
    Quillot, A.: Cyclic orders, European J. Combin. 10 (1989), 477–488.MathSciNetGoogle Scholar
  33. 33.
    Reinelt, G.: The Linear Ordering Problem: Algorithms and Applications, Res. Exp. Math. 8, Heldermann Verlag, Berlin, 1985.zbMATHGoogle Scholar
  34. 34.
    Spence, E.: Two-graphs, In: C. Colbourn and J. Dinitz (eds), The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996, pp. 686–695.Google Scholar
  35. 35.
    Trotter, W.: Combinatorics and Partially Ordered Sets, Dimension Theory, The Johns Hopkins University Press, Baltimore, MA, 1992.zbMATHGoogle Scholar
  36. 36.
    Trotter, W. and Moore, J.: Characterization problems for graphs, partially ordered sets, lattices, and families of sets, Discrete Math. 16 (1976), 361–381.MathSciNetCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Samuel Fiorini
    • 1
  • Peter C. Fishburn
    • 2
  1. 1.Département de Mathématique, CP 215Université Libre de Bruxelles, tBrusselsBelgium
  2. 2.AT&T Laboratories-ResearchFlorham ParkU.S.A.

Personalised recommendations