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In this article we give a new definition of direct product of two arbitraryfuzzy graphs. We define the concepts of domination and total domination in thisnew product graph. We obtain an upper bound for the total domination number ofthe product fuzzy graph. Further we define the concept of totalalpha-domination number and derive a lower bound for the total dominationnumber of the product fuzzy graph in terms of the total alpha-domination numberof the component graphs. A lower bound for the domination number of the samehas also been found.

Some results of domination and total domination in the direct product of two fuzzy graphs