Skip to main content

Threshold graphs under picture Dombi fuzzy information

Abstract

The aggregation feature, decision-making skills and operational characteristics of multi-purpose Dombi operators make them a highly adaptable tool for compiling the imprecise information. This study exploits the generalized structure of Dombi operators and significant characteristics of picture fuzzy sets \((\mathcal {PFS}_{s})\) to extend the theory of fuzzy graph by presenting the premium concept of picture Dombi fuzzy threshold graphs \((\mathcal {PDFTG}_{s}).\) We prove that \(\mathcal {PDFTG}_{s}\) do not induce picture Dombi fuzzy alternating \((\mathcal {PDFA})\) 4-cycle as induced subgraph, and these graphs can be constructed periodically by adding an isolated or dominant vertex to a single vertex graph. We demonstrate that \(\mathcal {PDFTG}_{s}\) are triangulated graphs. We show that the crisp graph of \(\mathcal {PDFTG}\) is a split graph \(({\mathcal {S}}{\mathcal {G}})\). Further, we illustrate the notion of threshold dimension and threshold partition number of picture Dombi fuzzy graphs \((\mathcal {PDFG}_{s})\). Moreover, we present some fundamental results related to threshold dimension and threshold partition number with the appropriate illustration. Finally, we discuss the implementation of \(\mathcal {PDFTG}_{s}\) in the distribution of coal resources.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

References

  • Akram M, Habib A (2019) \(q\)-rung picture fuzzy graphs: a creative view on regularity with applications. J Appl Math Comput 61:235–280

    MathSciNet  Article  Google Scholar 

  • Akram M, Khan A (2021) Complex pythagorean Dombi fuzzy graphs for decision making. Granul Comput 6:645–669

    MathSciNet  Article  Google Scholar 

  • Akram M, Shahzadi G (2021) Decision-making approach based on Pythagorean Dombi fuzzy soft graphs. Granul Comput 6:671–689

    Article  Google Scholar 

  • Akram M, Habib A, Koam AN (2019) A novel description on edge-regular q-rung picture fuzzy graphs with application. Symmetry 11(4):489

    Article  Google Scholar 

  • Akram M, Dudek W, Habib A, AlKenani A (2020a) Imperfect competition models in economic market structure with q-rung picture fuzzy information. J Intell Fuzzy Syst 38(4):5107–5126

    Article  Google Scholar 

  • Akram M, Dar JM, Naz S (2020b) Pythagorean Dombi fuzzy graphs. Compl Intell Syst 6:29–54

    Article  Google Scholar 

  • Akram M, Ahmad U, Rukhsar Karaaslan F (2021a) Complex Pythagorean fuzzy threshold graphs with application in petroleum replenishment. J Appl Math Comp. https://doi.org/10.1007/s12190-021-01604-y

    Article  MATH  Google Scholar 

  • Akram M, Habib A, Alcantud JCR (2021b) An optimization study based on Dijkstra algorithm for a network with picture trapezoidal fuzzy numbers. Neural Comput Appl 33:1329–1342

    Article  Google Scholar 

  • Akram M, Shahzadi G, Alcantud JCR (2021c) Multi-attribute decision-making with q-rung picture fuzzy information. Granul Comput. https://doi.org/10.1007/s41066-021-00260-8

    Article  Google Scholar 

  • Ali G, Akram M (2020) Decision-making method based on fuzzy \(N\)-soft expert sets. Arab J Sci Eng 45:10381–10400

    Article  Google Scholar 

  • Ali G, Ansari MN (2021) Multiattribute decision-making under Fermatean fuzzy bipolar soft framework. Granul Comput. https://doi.org/10.1007/s41066-021-00270-6

    Article  Google Scholar 

  • Ali G, Muhiuddin G, Adeel A, Abidin MZ (2021) Ranking effectiveness of COVID-19 tests using fuzzy bipolar soft expert sets. Math Prob Eng. https://doi.org/10.1155/2021/5874216

    Article  Google Scholar 

  • Alsina C, Trillas E, Valverde L (1983) On some logical connectives for fuzzy sets theory. J Math Anal Appl 93(1):15–26

    MathSciNet  Article  Google Scholar 

  • Andelic M, Simic SK (2010) Some notes on the threshold graphs. Discrete Math 310:2241–2248

    MathSciNet  Article  Google Scholar 

  • Ashraf S, Naz S, Kerre EE (2018) Dombi fuzzy graphs. Fuzzy Inf Eng 10(1):58–79

    Article  Google Scholar 

  • Atanassov KT (1986) Intuitionistic fuzzy sets. Physica 20(1):87–96

    MATH  Google Scholar 

  • Chen SM (1997) Interval-valued fuzzy hypergraph and fuzzy partition. IEEE Trans Syst Man Cybern Part B 27(4):725–733

    Article  Google Scholar 

  • Chen SM, Hsaio WH (2000) Bidirectional approximate reasoning for rule-based systems using interval-valued fuzzy sets. Fuzzy Sets Syst 113(2):185–203

    MathSciNet  Article  Google Scholar 

  • Chen J, Ye J (2017) Some single-valued neutrosophic Dombi weighted aggregation operators for multiple attribute decision making. Symmetry 9(6):82. https://doi.org/10.3390/sym9060082

    Article  Google Scholar 

  • Chen SM, Hsaio WH, Jong WT (1997) Bidirectional approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst 91(3):339–353

    MathSciNet  Article  Google Scholar 

  • Chvatal V, Hammer PL (1973) Set packing problems and threshold graphs. University of Waterloo, Waterloo, pp 21–73

    Google Scholar 

  • Cuong BC (2014) Picture fuzzy sets. J Comput Sci Cyber 30(4):409–420

    Google Scholar 

  • Cuong BC, Kreinovich V (2013) Picture fuzzy sets-a new concept for computational intelligence problems. In: Proceedings of the Third World Congress on Information and Communication Technologies (WICT’2013), Hanoi, Vietnam, pp 1–6

  • Dombi J (1982) A general class of fuzzy operators, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst 8(2):149–163

    MathSciNet  Article  Google Scholar 

  • Dubois D, Ostasiewicz W, Prade H (2000) Fuzzy sets: history and basic notions. Handbook of fuzzy sets and possibility theory. Springer, New York, pp 121–124

    Book  Google Scholar 

  • Hamacher H (1978) On logical aggregations of non-binar explicit decision criteria. Fischer Verlag, Frankfurt

    Google Scholar 

  • Hameed S, Akram M, Mustafa N, Karaaslan F (2021) Extension of threshold graphs under complex intuitionistic fuzzy environment. J Mult-Valued Logic Soft Comput 37:295–315

    MATH  Google Scholar 

  • Henderson PB, Zalcstein Y (1977) A graph-theoretic characterization of the PV class of synchronizing primitives. SIAM J Comput 6(1):88–108

    MathSciNet  Article  Google Scholar 

  • Jana C, Senapati T, Pal M, Yager RR (2019a) Picture fuzzy Dombi aggregation operators: application to MADM process. Appl Soft Comput 74:99–109

    Article  Google Scholar 

  • Jana C, Pal M, Wang J (2019b) Bipolar fuzzy Dombi aggregation operators and its application in multiple-attribute decision-making process. J Amb Intell Hum Comput 10:3533–3549

    Article  Google Scholar 

  • Klement PE, Mesiar R, Pap E (2000) Triangular norms, vol. 8 of trends in logic-studia logica library. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  • Koop GJ (1986) Cyclic scheduling of offweekends. Oper Res Lett 4:259–263

    MathSciNet  Article  Google Scholar 

  • Kuwagaki A (1952) On the rational functional equation of function unknown of two variables. Mem Coll Sci 28(2)

  • Liu P, Shahzadi G, Akram M (2020) Specific types of q-rung picture fuzzy Yager aggregation operators for decision-making. Int J Comput Intell Syst 13(1):1072–1091

    Article  Google Scholar 

  • Mahapatra T, Pal M (2021) An investigation on m-polar fuzzy threshold graph and its application on resource power controlling system. J Amb Intell Hum Comput. https://doi.org/10.1007/s12652-021-02914-6

    Article  Google Scholar 

  • Menger K (1942) Statistical metrics. Proc Natl Acad Sci USA 28(12):535–537

    MathSciNet  Article  Google Scholar 

  • Mohanta K, Dey A, Pal A (2020) A study on picture Dombi fuzzy graph. Decis Mak Appl Manag Eng 3(2):119–130

    Article  Google Scholar 

  • Mordeson JN, Nair PS (2001) Fuzzy graphs and fuzzy hypergraphs, 2nd edn. Physica Verlag, Heidelberg

    MATH  Google Scholar 

  • Mordeson JN, Peng CS (1994) Operations on fuzzy graphs. Inf Sci 79(3–4):159–170

    MathSciNet  Article  Google Scholar 

  • Naz S, Ashraf S, Akram M (2018) A novel approach to decision-making with Pythagorean fuzzy information. Mathematics 6:1–28

    Article  Google Scholar 

  • Ordman ET (1985) Threshold coverings and resource allocation. In:16th Southeastern Conference on Combinatorics, Graph Theory and Computing, pp 99–113

  • Peled UN, Mahadev NV (1995) Threshold graphs and retaed topics, vol 56. North Holland, pp 1–543

  • Pramanik T, Pal M, Mondal S (2016) Intervel-valued fuzzy threshold graph. Pac Sci Rev A 18(1):66–71

    Google Scholar 

  • Rosenfeld A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M (eds) Fuzzy sets and their applications to cognitive and decision process. Academic Press, London, pp 77–95

    Chapter  Google Scholar 

  • Samanta S, Pal M (2011) Fuzzy threshold graphs. Int J Fuzzy Syst 3(12):360–364

    Google Scholar 

  • Schweizer B, Sklar A (1960) Statistical metric spaces. Pac J Math 10(1):313–334

    MathSciNet  Article  Google Scholar 

  • Shannon A, Atanassov KT (1994) A first step to a theory of intuitionistic fuzzy graphs. In: Proceedings of Fuzzy Based Expert Systems, D. Lakov, Ed, Sofia, pp 59–61

  • Shi L, Ye J (2018) Dombi aggregation operators of neutrosophic cubic sets for multiple attribute decision-making. Algorithms. https://doi.org/10.3390/a11030029

    Article  Google Scholar 

  • Sittara M, Akram M, Riaz M (2021) Decision-making analysis based on q-rung picture fuzzy graph structures. J Appl Math Comput. https://doi.org/10.1007/s12190-020-01471-z

    MathSciNet  Article  MATH  Google Scholar 

  • Turksen IB (1986) Interval valued fuzzy sets based on normal forms. Fuzzy Sets Syst 20(2):191–210

    MathSciNet  Article  Google Scholar 

  • Wei G (2017) Some cosine similarity measures for picture fuzzy sets and their applications to strategic decision making. Informatica 28(3):547–564

    Article  Google Scholar 

  • Yager RR (2013) Pythagorean fuzzy subsets. In: IEEE, pp 57–61

  • Yang L, Mao H (2019) Intuitionistic fuzzy threshold graphs. J Intell Fuzzy Syst 36:6641–6651

    Article  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  Google Scholar 

  • Zuo C, Pal A, Dey A (2019) New concepts of picture fuzzy graphs with application. Mathematics. https://doi.org/10.3390/math7050470

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Muhammad Akram.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest regarding the publication of this article.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Akram, M., Ahmad, U. & Rukhsar Threshold graphs under picture Dombi fuzzy information. Granul. Comput. 7, 691–707 (2022). https://doi.org/10.1007/s41066-021-00291-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41066-021-00291-1

Keywords

  • Dombi operators
  • 4-cycle
  • Picture Dombi fuzzy threshold dimension
  • Partition number