Hypergraphs for Interval-Valued Structures

Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 390)


In this chapter, we present interval-valued fuzzy hypergraphs, \(A=[\mu ^-, \mu ^+]\)–tempered interval-valued fuzzy hypergraphs, and some of their properties. Moreover, we discuss the notions of vague hypergraphs, dual vague hypergraphs, and A-tempered vague hypergraphs. Finally, we describe interval-valued intuitionistic fuzzy hypergraphs and interval-valued intuitionistic fuzzy transversals of \(\mathscr {H}\). This chapter is due to [4, 5, 6, 11, 22, 25].


  1. 1.
    Akram, M., Dudek, W.A.: Intuitionistic fuzzy hypergraphs with applications. Inf. Sci. 218, 182–193 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Akram, M., Dudek, W.A.: Interval-valued fuzzy graphs. Comput. Math. Appl. 61, 289–299 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Akram, M., Feng, F., Sarwar, S., Jun, Y.B.: Certain types of vague graphs. UPB Scientific Bulletin, Series \(A\)–Applied Mathematics and Physics, vol. 3, pp. 1–15 (2013)Google Scholar
  4. 4.
    Akram, M., Alshehri, N.O.: Tempered interval-valued fuzzy hypergraphs. Scientific Bulletin Series A–Applied Mathematics and Physics, vol. 77(1), pp. 39–48 (2015)Google Scholar
  5. 5.
    Akram, M., Gani, N., Saeid, A.B.: Vague hypergraphs. J. Intell. Fuzzy Syst. 26, 647–653 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Atanassov, K.T., Gargov, G.: Interval-valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 31(3), 343–349 (1989)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Berge, C.: Graphs and Hypergraphs. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  8. 8.
    Chen, S.M.: Interval-valued fuzzy hypergraph and fuzzy partition. IEEE Trans. Syst. Man Cybern. (Cybernetics) 27(4), 725–733 (1997)Google Scholar
  9. 9.
    Deschrijver, G., Cornelis, C.: Representability in interval-valued fuzzy set theory. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 15, 345–361 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Deschrijver, G., Kerre, E.E.: On the relationships between some extensions of fuzzy set theory. Fuzzy Sets Syst. 133, 227–235 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Trans. Syst. Man Cybern. 23, 610–614 (1993)CrossRefGoogle Scholar
  12. 12.
    Goetschel Jr., R.H.: Introduction to fuzzy hypergraphs and Hebbian structures. Fuzzy Sets Syst. 76, 113–130 (1995)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Goetschel Jr., R.H., Craine, W.L., Voxman, W.: Fuzzy transversals of fuzzy hypergraphs. Fuzzy sets Syst. 84, 235–254 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gorzalczany, M.B.: A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst. 21, 1–17 (1987)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gorzalczany, M.B.: An Interval-valued fuzzy inference method some basic properties. Fuzzy Sets Syst. 31, 243–251 (1989)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hongmei, J., Lianhua, W.: Interval-valued fuzzy subsemigroups and subgroups associated by interval-valued fuzzy graphs. In: 2009 WRI Global Congress on Intelligent Systems, pp. 484–487 (2009)Google Scholar
  17. 17.
    Kaufmann, A.: Introduction a la Thiorie des Sous-Ensemble Flous, vol. 1. Masson, Paris (1977)Google Scholar
  18. 18.
    Lee, K.M.: Comparison of interval-valued fuzzy sets, intuitionistic fuzzy sets, and bipolar-valued fuzzy sets. J. Fuzzy Log. Intell. Sys. 14, 125–129 (2004)Google Scholar
  19. 19.
    Lee-kwang, H., Lee, K.-M.: Fuzzy hypergraph and fuzzy partition. IEEE Trans. Syst. Man Cybern. 25(1), 196–201 (1995)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mendel, J.M.: Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Prentice-Hall, Upper Saddle River, NJ (2001)zbMATHGoogle Scholar
  21. 21.
    Mordeson, J.N., Nair, P.S.: Fuzzy Graphs and Fuzzy Hypergraphs, 2nd edn. Physica Verlag, Heidelberg (2001)zbMATHGoogle Scholar
  22. 22.
    Naz, S., Malik, M.A., Rashmanlou, H.: Hypergraphs and transversals of hypergraphs in interval-valued intuitionistic fuzzy setting. J. Mult.-Valued Log. Soft Comput. 30, 399–417 (2018)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Roy, M.K., Biswas, R.: l-v fuzzy relations and Sanchez’s approach for medical diagnosis. Fuzzy Sets Syst. 47, 35–38 (1992)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Turksen, I.B.: Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst. 20, 191–210 (1986)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefGoogle Scholar
  26. 26.
    Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3(2), 177–200 (1971)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zadeh, L.A.: The concept of a linguistic and application to approximate reasoning-I. Inf. Sci. 8, 199–249 (1975)Google Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of the PunjabLahorePakistan
  2. 2.Department of MathematicsUniversity of the PunjabLahorePakistan

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