Topological Ordering on Interval Type-2 Fuzzy Graph

  • Margarita KnyazevaEmail author
  • Stanislav BelyakovEmail author
  • Janusz KacprzykEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 896)


The topological ordering for graphs has many practical applications where the nature of the stated problem requires sequential processing. It might be linking and loading problems, planning and scheduling algorithms, assembly line processing and many other practical applications where precedence constraints are met. Sequencing of vertices execution usually depends on problem and can be represented by directed acyclic graph structure with expert estimations of uncertain variables one can come across. Difficulties in ordering and scheduling vertices of such fuzzy-estimated weighted graph are investigated in this paper. An algorithm for topological ordering and directed minimum spanning tree problem of interval type-2 fuzzy graph for scheduling problem is developed.


Fuzzy graph Scheduling Decision-making Type-2 fuzzy numbers 



This work has been supported by the Ministry of Education and Science of the Russian Federation under Project “Methods and means of decision making on base of dynamic geographic information models” (Project part, State task 2.918.2017), and the Russian Foundation for Basic Research, Project № 18-01-00023a.


  1. 1.
    Rosenfeld, A.: Fuzzy graphs. In: Zadeh, L.A., Fu, K.S., Shimura, M. (eds.) Fuzzy Sets and Their Applications, pp. 77–95. Academic Press, New York (1975)Google Scholar
  2. 2.
    Li, F., Zang, L., Zang, Z.: Dynamic Fuzzy Machine Learning, pp. 251–252. Walter De Gruyter GmbH, Berlin/Boston (2018)Google Scholar
  3. 3.
    Mordeson, J.N., et al. (eds.): Fuzzy Graphs and Fuzzy Hypergraphs. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  4. 4.
    Mathew, S., Mordeson, J.N., Malik, D.S.: Fuzzy Graph Theory. Studies in Fuzziness and Soft Computing, pp. 19–20. Springer, Cham (2018)CrossRefGoogle Scholar
  5. 5.
    Knyazeva, M., Bozhenyuk, A., Rozenberg, I.: Scheduling alternatives with respect to fuzzy and preference modeling on time parameters. In: Proceedings of the 10th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2017), Warsaw, Poland, 11–15 September 2017. Advances in Intelligent Systems and Computing Book Series (AISC), vol. 642, pp. 358–369. Springer, Cham (2017)Google Scholar
  6. 6.
    Mendel, J.M., John, R.I.: Type-2 fuzzy sets made simple. IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002)CrossRefGoogle Scholar
  7. 7.
    Miller, S., Wagner, C., Garibaldi, J.M., Appleby, S.: Constructing general Type-2 fuzzy sets from interval-valued data. In: Proceedings of WCCI 2012 IEEE World Congress on Computational Intelligence (2012)Google Scholar
  8. 8.
    Choi, Myeong-Bok, Lee, Sang-Un: An efficient implementation of Kruskal’s and reverse-delete minimum spanning tree algorithm. J. Inst. Internet Broadcast. Commun. 13(2), 103–114 (2013)CrossRefGoogle Scholar
  9. 9.
    Choi, Myeong-Bok, Lee, Sang-Un: A prim minimum spanning tree algorithm for directed graph. J. Inst. Internet Broadcast. Commun. 12(3), 51–61 (2012)CrossRefGoogle Scholar
  10. 10.
    Edmonds, J.: Optimum branchings. J. Res. Nat. Bur. Stand. 71B, 233–240 (1967)MathSciNetCrossRefGoogle Scholar
  11. 11.
    An open source implementation of Edmonds’s algorithm written in C ++ and licensed under the MIT License:

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Southern Federal UniversityTaganrogRussia
  2. 2.Systems Research Institute, Polish Academy of SciencesWarsawPoland

Personalised recommendations