Picture fuzzy tolerance graphs with application

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.