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.

In this paper, bipolar fuzzy planar graph is defined and studied several properties. The bipolar fuzzy planar graph is defined in a very interesting way. The parameter “degree of planarity” measures the planarity of a bipolar fuzzy graph. The other relevant terms such as strong edges, bipolar fuzzy faces, strong bipolar fuzzy faces are defined here. A very close association of bipolar fuzzy planar graph is bipolar fuzzy dual graph. This is also defined and several properties of it are studied. Bipolar fuzzy planar graph and bipolar fuzzy dual graph have many applications in different fields including design of social network, design of subway tunnel or routes, gas or oil pipelines, image segmentation, etc.