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In this paper, fuzzy $$\phi $$ -tolerance competition graphs are defined. Several interesting properties are investigated. Here $$\phi $$ is any real valued function. Particular cases of $$\phi $$ , viz. minimum, maximum and sum are considered. A daily life problem is modelled using fuzzy min-tolerance competition graph. Few properties are investigated for fuzzy tolerance graphs. [ABSTRACT FROM AUTHOR]

This paper develops an interval-valued fuzzy $$\phi$$ -tolerance competition graphs which is the extension of basic fuzzy graphs and $$\phi$$ is any real valued function. Interval-valued fuzzy $$\phi$$ -tolerance competition graph is constructed by taking all the fuzzy sets of a fuzzy $$\phi$$ -tolerance competition graph as interval-valued fuzzy sets. Product of two IVFPTCGs and relations between them are defined. Here, some hereditary properties of products of interval-valued fuzzy $$\phi$$ -tolerance competition graphs are represented. Application of interval-valued fuzzy competition graph in image matching is given to illustrate the model. [ABSTRACT FROM AUTHOR]