Abstract
In this study, the notions of picture fuzzy tolerance graphs, picture fuzzy interval containment graphs and picture fuzzy phi -tolerance graphs are established. Three special types of picture fuzzy tolerance graphs having bounded representations are introduced and studied corresponding properties of them taking phi as max, min and sum functions. Also, picture fuzzy proper and unit tolerance graphs are established and some related results are investigated. The class of picture fuzzy phi -tolerance chaingraphs which is the picture fuzzy phi -tolerance graphs of a nested family of picture fuzzy intervals are presented. A real-life application in sports competition is modeled using picture fuzzy min-tolerance graph. Also a comparison is given between picture fuzzy tolerance graphs and intuitionistic fuzzy tolerance graphs.
Highlights
Research backgroundIn 1965, the notion of fuzzy set (FS) was initially posed by Zadeh [51] to model the problems having uncertainties
We have presented a new classification of tolerance graphs (TGs) in picture fuzzy (PF) environment
We present the innovative concept of PF tolerance graph (PFTG)
Summary
In 1965, the notion of fuzzy set (FS) was initially posed by Zadeh [51] to model the problems having uncertainties. Naz et al [26] introduced the novel concepts of energy of a graph in the context of a bipolar fuzzy environment with its application in decision making problem. Ghorai and Pal introduced some operations on m-polar FGs [15], some isomorphic properties of m-polar FGs with applications [16] and m-polar FPGs [17] They studied certain types of product bipolar FGs in [18]. Das and Ghorai [8] have defined picture FPGs and applied it to construct road map designs They have applied the idea of PF sets to CGs, PF genus graphs [9,10,11]. The PFTG models give more legibility, flexibility and suitability to the system as compared with the models in other fields due to uncertainty
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