Document Type : Research Paper


1 Department of Mathematics, St. Joseph's Institute of Technology, OMR, Chennai, India.

2 Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, Chennai, India.

3 Laboratory of Information Processing, Faculty of Science Ben M'Sik, University of Hassan II, Casablanca, Morocco.

4 Department of Mathematics, Science College, Yarmouk University, Irbid-Jordan.


The notion of Fuzzy Graphs (FGs) and Intuitionistic Fuzzy Graphs (IFGs) is generalized in the Picture Fuzzy Graph (PFG), which is a more common platform for expressing the degree of positive, negative, and neutral membership functions. Picture Fuzzy Soft Graphs (PFSGs) are powerful mathematical tools for modeling real-world vagueness. The concept of vulnerability parameters of PFSG is provided in this research work by introducing the novel cardinality, domination number, integrity, and Domination Integrity (ÐÏ) of PFSG. Furthermore, a decision-making method for the PFSG has been presented using ÐÏ to suggest an algorithm for determining the ideal location for establishing a city diagnosis center


Main Subjects

[1]     Zadeh, L. A. (1965). Fuzzy sets. Information and control, 8(3), 338–353. DOI:10.1016/S0019-9958(65)90241-X
[2]     Molodtsov, D. (1999). Soft set theory—first results. Computers & mathematics with applications, 37(4–5), 19–31.
[3]     Atanassov, K. T. (1986). Intuitionistic fuzzy sets. Fuzzy sets and systems, 20(1), 87–96. DOI:10.1016/S0165-0114(86)80034-3
[4]     Cuong, B. C., & Kreinovich, V. (2014). Picture fuzzy sets - a new concept for computational intelligence problems. 2013 3rd world congress on information and communication technologies, wict 2013 (pp. 1–6). IEEE.
[5]     Thumbakara, R. K., & George, B. (2014). Soft graphs. General mathematics notes, 21(2), 75–86.
[6]     Euler, L. (1741). Solutio problematis ad geometriam situs pertinentis. Commentarii academiae scientiarum petropolitanae, 8, 128–140.
[7]     Rosenfeld, A. (1975). Fuzzy graphs. In Fuzzy sets and their applications to cognitive and decision processes (pp. 77–95). Academic Press.
[8]     Akram, M., & Nawaz, S. (2016). Fuzzy soft graphs with applications. Journal of intelligent and fuzzy systems, 30(6), 3619–3632. DOI:10.3233/IFS-162107
[9]     Atanassov, K. T., & Shannon, A. (1994). A first step to a theory of the intuitionistic fuzzy graphs [presentation]. Proceedings of the first workshop on fuzzy based expert systems, sofia (pp. 59–61).
[10]   Shyla, A. M., & Varkey, T. K. M. (2016). Intuitionistic fuzzy soft graph. International journal of fuzzy mathematical archive, 11(02), 63–77. DOI:10.22457/ijfma.v11n2a2
[11]   Smarandache, F. (1998). Neutrosophy: neutrosophic probability, set, and logic: analytic synthesis & synthetic analysis. Rehoboth: American research press, 2020.
[12]   Broumi, S., Talea, M., Bakali, A., & Smarandache, F. (2016). Single valued neutrosophic graphs. Journal of new theory, 10, 86–101.
[13]   Zuo, C., Pal, A., & Dey, A. (2019). New concepts of picture fuzzy graphs with application. Mathematics, 7(5), 470. DOI:10.3390/math7050470
[14]   Mohamedlsmayil, A., & AshaBosely, N. (2019). Domination in picture fuzzy graphs. American international journal of research in science, technology, engineering & mathematics, 205–210.
[15]   Chellamani, P., Ajay, D., Broumi, S., & Ligori, T. A. A. (2022). An approach to decision-making via picture fuzzy soft graphs. Granular computing, 7(3), 527–548. DOI:10.1007/s41066-021-00282-2
[16]   Barefoot, C. A., Entringer, R., & Swart, H. C. (1987). Vulnerability in graphs - a comparative survey. Journal combination mathematical combin computer, 1(38), 13–22.
[17]   Sundareswaran, R., & Swaminathan, V. (2016). Integrity and domination integrity of gear graphs. TWMS journal of applied and engineering mathematics, 6(1), 54–63.
[18]   Saravanan, M., Sujatha, R., & Sundareswaran, R. (2016). Integrity of fuzzy graphs. Bulletin of the international mathematical virtual institute6, 89-96.
[19]   Ganesan, B., Raman, S., & Pal, M. (2022). Strong domination integrity in graphs and fuzzy graphs. Journal of intelligent and fuzzy systems, 43(3), 2619–2632. DOI:10.3233/JIFS-213189
[20]   Jaikumar, R. V., Sundareswaran, R., & Broumi, S. (2023). Integrity and domination integrity in neutrosophic soft graphs. Neutrosophic sets and systems, 53, 165–178. DOI:10.5281/zenodo.7535995