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The Fixed Point Property for Ordered Sets of Interval Dimension 2

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Abstract

We provide a polynomial time algorithm that identifies if a given finite ordered set is in the class of d2-collapsible ordered sets. For a d2-collapsible ordered set, the algorithm also determines if the ordered set is connectedly collapsible. Because finite ordered sets of interval dimension 2 are d2-collapsible, in particular, the algorithm determines in polynomial time if a given finite ordered set of interval dimension 2 has the fixed point property. This result is also a first step in investigating the complexity status of the question whether a given collapsible ordered set has the fixed point property.

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References

  1. Duffus, D., Goddard, T.: The complexity of the fixed point property. Order 13, 209–218 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Duffus, D., Poguntke, W., Rival, I.: Retracts and the fixed point problem for finite partially ordered sets. Canad. Math. Bull. 23, 231–236 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Farley, J.D.: The uniqueness of the core. Order 10, 129–131 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fofanova, T., Rutkowski, A.: The fixed point property in ordered sets of width two. Order 4, 101–106 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Felsner, S.: 3-Interval irreducible partially ordered sets. Order 11, 97–125 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fofanova, T., Rival, I., Rutkowski, A.: Dimension 2, fixed points and dismantlable ordered sets. Order 13, 245–253 (1994)

    MathSciNet  MATH  Google Scholar 

  7. Rival, I.: A fixed point theorem for finite partially ordered sets. J. Comb. Theory (A) 21, 309–318 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rutkowski, A.: The fixed point property for small sets. Order 6, 1–14 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Schröder, B.: Fixed point property for 11-element sets. Order 10, 329–347 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  10. Schröder, B.: Fixed Cliques and Generalizations of Dismantlability. In: Alavi, Y. et al. (eds.) Combinatorics, Graph Theory, and Algorithms, Proceedings of the Eighth Quadrennial International Conference in Graph Theory, Combinatorics, Algorithms and Applications, vol. II, pp. 747–756 (1996)

  11. Schröder, B.: The uniqueness of cores for chain-complete ordered sets. Order 17, 207–214 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Schröder, B.: Ordered Sets – an Introduction with Connections from Combinatorics to Topology. Birkhäuser, Boston (2016)

    MATH  Google Scholar 

  13. Schröder, B.: The use of retractions in the fixed point theory for ordered sets. In: Alfuraidan, M., Ansari, Q.H. (eds.) Fixed Point Theory and Graph Theory – Foundations and Integrative Approaches, ElSevier, pp. 365–417 (2016). (Proceedings of the Workshop on Fixed Point Theory and Applications, King Fahd University of Petroleum and Minerals, December 2014, Dhahran, Saudi Arabia.)

  14. Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Alg. Disc. Meth. 3, 351–358 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, The University of Southern Mississippi, 118 College Avenue, #5045, Hattiesburg, MS 39406, USA

    Bernd S. W. Schröder

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  1. Bernd S. W. Schröder
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Correspondence to Bernd S. W. Schröder.

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Schröder, B.S.W. The Fixed Point Property for Ordered Sets of Interval Dimension 2. Order 34, 307–322 (2017). https://doi.org/10.1007/s11083-016-9401-4

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  • DOI: https://doi.org/10.1007/s11083-016-9401-4

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