Fuzzy permutation graph and its complements

This paper deals with fuzzy permutation graph which is introduced as a fuzzified model of a crisp permutation graph. We have defined two types of complements of a fuzzy permutation graph viz. p-complement and f-complement. We have considered various examples to encounter the existence and non-existence of the f-complement graph as a fuzzy permutation graph and established a sufficient condition for the existence of the same. Also, we have calculated the possible number of fuzzy permutation graphs that can represent the f-complement of a fuzzy permutation graph for a given permutation depending on the edge-membership values of the fuzzy permutation graph. We have also introduced and characterized fuzzy comparability graph. Lastly, we have discussed a real-life application of fuzzy permutation graph.