Decomposition of realizable fuzzy relations

This paper deals with the decomposition problem of realizable fuzzy relations. First, given a realizable fuzzy relation $$A$$ , a method to construct a fuzzy relation $$B$$ such that $$A=B\odot B^T$$ (where $$\odot $$ is the max---min composition, $$B^T$$ denotes the transpose of $$B$$ ) is proposed. Then it is proved that the content of a realizable fuzzy relation is equal to the chromatic number of a simple graph generated by the realizable fuzzy relation. Therefore, many existing algorithms (including exact and heuristic algorithms) developed to find the chromatic number or to get an upper bound on chromatic number of a graph can be applied to solve the calculating problem of the content of a realizable fuzzy relation.