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Cybernetics and Systems Analysis

, Volume 55, Issue 2, pp 313–320 | Cite as

Representation of Fragmentary Structures by Oriented Graphs

  • O. V. KryvtsunEmail author
Article
  • 7 Downloads

Abstract

This paper investigates properties of fragmentary structures and establishes a relation between them and marked acyclic digraphs with one source and also a correspondence between classes of isomorphic fragmentary structures and unmarked acyclic digraphs of certain type, which are called feasible graphs. The concepts of a dimension of a feasible graph and its corresponding isomorphic fragmentary structures are defined. An expression is obtained for the lower-bound estimate of a dimension. A theorem on properties of feasible graphs is proved. The numbers of fragmentary structures and classes of isomorphic fragmentary structures of small dimensions are counted.

Keywords

fragmentary structure partially ordered set directed acyclic graph hypercube 

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References

  1. 1.
    H. Whitney, “On the abstract properties of linear dependence,” American Journal of Mathematics, Vol. 57, No. 3, 509–533 (1935).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Bjorner and G. M. Ziegler, “Introduction to greedoids,” in: N. White (ed.), Matroid Applications, Cambridge University Press (1992), pp. 284–357.Google Scholar
  3. 3.
    V. P. Ilyev, “Problems on independence systems solvable by the greedy algorithm,” Discrete Math. Appl., Vol. 19, Iss. 5, 512–522 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    I. V. Kozin, O. V. Kryvtsun, and V. P. Pinchuk, “Evolutionary-fragmentary model of the routing problem,” Cybernetics and Systems Analysis, Vol. 51, No. 3, 432–437 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    I. V. Kozin, O. V. Kryvtsun, and S. I. Poluga, “Fragmentary structure and evolutionary algorithm for rectangular cutting problems,” Visnyk of Zaporizhzhya National University, Ser. Physical & Mathematical Sciences, No. 2, 65–72 (2014).Google Scholar
  6. 6.
    I. V. Kozin and O. V. Kryvtsun, “Modelling of the single-layer and two-layer routing problems,” USiM, No. 2, 58–64 (2016).Google Scholar
  7. 7.
    I. V. Kozin, S. Yu. Borue, and O. V. Kryvtsun, “Mathematical model of different type bin packing,” Visnyk of Zaporizhzhya National University, Ser. Economical Sciences, No. 2, 85–92 (2016).Google Scholar
  8. 8.
    Ye. V. Kryvtsun, “Evolutionary-fragmentary algorithm of finding the minimal axiom set,” USiM, No. 5, 25–31 (2016).Google Scholar
  9. 9.
    I. V. Kozin and S. I. Polyuga, “Properties of fragmentary structures,” Visnyk of Zaporizhzhya National University, Ser. Mathematical Modeling and Applied Mechanics, No. 1, 99–106 (2012).Google Scholar
  10. 10.
    S. I. Poluga, Fragmental Optimization Models in Problems of Covering Graphs with Typical Subgraphs [in Ukrainian], PhD Thesis in Phys.-math. Sciences, Zaporizhzhya (2015). URL: http://phd.znu.edu.ua/page/dis/06_2016/Polyuga_dis.pdf.
  11. 11.
    C. Berge, Théorie des Graphes et ses Applications [Russian translation], Izd. Innostr. Lit., Moscow (1962).Google Scholar
  12. 12.
    F. Harary and E. M. Palmer, Graphical Enumeration [Russian translation], Mir, Moscow (1977).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Zaporizhzhya National Technical UniversityZaporizhzhyaUkraine

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