In this paper newly introduced the definition of α-Cut diminish fuzzy graph and extended the new concept “α-Cut strong arc” of α-Cut diminish fuzzy graph. Further definition of domination in α-Cut diminish fuzzy graph by α-Cut strong arc is newly introduced. Moreover, some standard theorems and related result are discussed with suitable example of Standard α-Cut diminish fuzzy graph like fuzzy path FPn, fuzzy cycle FCn and fuzzy complete graph FKn. Further complement of α-Cut diminish fuzzy graph are introduced and discussed with new application.

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