Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The jump number of Z-free ordered sets

  • 22 Accesses

  • 2 Citations

Abstract

An ordered set P is called Z-free if it does not contain a four-element subset {a, b, c, d} such that a<b and c<b are the only comparabilities among these elements. In this paper we present a polynomial algorithm to find the jump number of finite Z-free ordered sets and that of their duals.

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

References

  1. 1.

    M.Chein and M.Habib (1984) Jump number of dags having Dilworth number 2, Discrete Applied Math. 7, 243–250.

  2. 2.

    O.Cogis and M.Habib (1979) Nombre de sauts et graphes series-paralleles, RAIRO Inf. Th. 13 (1), 3–18.

  3. 3.

    C. J.Colbourn and W. R.Pulleyblank (1985) Minimizing setups in ordered sets of fixed width, Order 1, 225–229.

  4. 4.

    D.Duffus, I.Rival and P.Winkler (1982) Minimizing setups for cycle-free orders sets, Proc. of the American Math. Soc. 85(4), 509–513.

  5. 5.

    U.Faigle and R.Schrader (1985) A setup heuristic for interval orders, Oper. Res. Letters 4, 185–188.

  6. 6.

    W. R. Pulleyblank (1981) On minimizing setups in precedence constrained scheduling, Report 81105-OR, University of Bonn.

  7. 7.

    I.Rival (1983) Optimal linear extensions by interchanging chains, Proc. Amer. Math. Soc. 83, 387–394.

  8. 8.

    I. Rival (1989) Problem 2.2, in Algorithms and Other (I. Rival, ed.), Kluwer Acad. Publ., p. 475.

  9. 9.

    I.Rival and N.Zaguia (1986) Constructing greedy linear extensions by interchanging chains, Order 3, 107–121.

  10. 10.

    A. Sharary and N. Zaguia (1989) On a setup optimization problem for interval orders, preprint.

  11. 11.

    A.Sharary and N.Zaguia (1991) On minimizing jumps for ordered sets, Order 7, 353–359.

  12. 12.

    G.Steiner and L. K.Stewart (1987) A linear algorithm to find the jump number of 2-dimensional bipartite partial orders, Order 3, 359–367.

  13. 13.

    N. Zaguia (1985) Schedules, cutsets and ordered sets, Ph.D. Thesis, University of Calgary.

Download references

Author information

Additional information

Communicated by I. Rival

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sharary, A. The jump number of Z-free ordered sets. Order 8, 267–273 (1991). https://doi.org/10.1007/BF00383447

Download citation

AMS subject classifications (1991)

  • Primary: 06A06
  • secondary: 68C25

Key words

  • Ordered set
  • linear extension
  • jump number