Analysis of Road Map Design Based on Multigraph with Picture Fuzzy Information


In this paper, the picture fuzzy planar graph is defined along with some basic properties of picture fuzzy graph. The notion of picture fuzzy planar graphs is introduced and the terms such as strong (weak) edges, strong (weak) picture fuzzy planar graphs, strength of an edge, degree of planarity, picture fuzzy faces, strong (weak) picture fuzzy faces, picture fuzzy dual graphs and picture fuzzy combinatorial dual graphs are defined. Some theorems on degree of planarity and picture fuzzy combinatorial dual graphs have been established. An application of picture fuzzy planar graph in the design of road maps is given.

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  1. 1.

    Al-Hawary, T., Mahamood, T., Jan, N., Ullah, K., Hussain, A.: On intuitionistic fuzzy graphs and some operations on picture fuzzy planar graphs. Ital. J. Pure Appl. Math. (2018)

  2. 2.

    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20(1), 87–96 (1986)

    MATH  Google Scholar 

  3. 3.

    Bhutani, K.R., Rosenfeld, A.: Strong arcs fuzzy graphs. Inf. Sci. 152(1), 319–322 (2003)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Cuong, B.C.: Picture fuzzy sets. J. Comput. Sci. Cybern. 30(4), 409–420 (2014)

    Google Scholar 

  5. 5.

    Cuong, B.C., Hai, P.V.: Some fuzzy logic operators for picture fuzzy sets. In: Proceedings of Seventh International Conference on Knowledge and Systems Engineering, pp. 132–137 (2015)

  6. 6.

    Ghorai, G., Pal, M.: A study on \(m\)-polar fuzzy planar graphs. Int. J. Comput. Sci. Math. 7(3), 283–292 (2016)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Nirmala, G., Dhanabal, K.: Special planar fuzzy graph configurations. Int. J. Sci. Res. Publ. 2(7), 1–4 (2012)

    Google Scholar 

  8. 8.

    Nagoorgani, A., Hussain, R.J.: Fuzzy effective distance K-dominating sets and their applications. Int. J. Algorithms Comput. Math. 2(3), 25–36 (2009)

    Google Scholar 

  9. 9.

    Poulik, S., Ghorai, G.: Detour \(g\)-interior nodes and detour \(g\)-boundary nodes in bipolar fuzzy graph with applications. Hacet. J. Math. Stat. 49(1), 106–119 (2020)

    Google Scholar 

  10. 10.

    Poulik, S., Ghorai, G.: Certain indices of graphs under bipolar fuzzy environment with applications. Soft Comput. 24, 5119–5131 (2020)

    Google Scholar 

  11. 11.

    Poulik, S., Ghorai, G.: Note on “Bipolar fuzzy graphs with applications”. Knowl. Based Syst. 192, 1–5 (2020)

    Google Scholar 

  12. 12.

    Pramanik, T., Samanta, S., Pal, M.: Interval-valued fuzzy planar graphs. Int. J. Mach. Learn. Cybern. 7(4), 653–664 (2014)

    Google Scholar 

  13. 13.

    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 

  14. 14.

    Samanta, S., Pal, A., Pal, M.: Concept of fuzzy planar graphs. In: Proceedings of the Science and Information Conference, London, UK, October 7–9, pp. 557–563 (2013)

  15. 15.

    Samanta, S., Pal, A., Pal, M.: New Concept of fuzzy planar graph. Int. J. Adv. Res. Artif. Intell. 3(1), 52–59 (2014)

    Google Scholar 

  16. 16.

    Samanta, S., Pal, M.: Fuzzy planar graphs. IEEE Trans. Fuzzy Syst. 23(6), 1936–1942 (2015)

    Google Scholar 

  17. 17.

    Shriram, S., Sujatha, R., Narasimman, S.: Fuzzy combinatorial dual graph. Int. J. Pure Appl. Math. 109(7), 35–41 (2016)

    Google Scholar 

  18. 18.

    Yahya Mohamed, S., Subashini, N.: Bipolar fuzzy planar graphs. Int. J. Appl. Eng. Res. 13(12), 10535–10538 (2018)

    Google Scholar 

  19. 19.

    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    MATH  Google Scholar 

Download references


The authors are highly grateful to the learned reviewers for their valuable comments and suggestions to improve the quality of the article.

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Correspondence to Ganesh Ghorai.

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Das, S., Ghorai, G. Analysis of Road Map Design Based on Multigraph with Picture Fuzzy Information. Int. J. Appl. Comput. Math 6, 57 (2020).

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  • Picture fuzzy graphs
  • Picture fuzzy planar graphs
  • Strong (weak) picture fuzzy planar graphs
  • Picture fuzzy dual graphs
  • Picture fuzzy combinatorial dual graph