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Let $(X,M,*)$ be a fuzzy metric space endowed with a graph $G$ such that the set $V\left(G\right)$ of vertices of $G$ coincides with $X$. Then we define a $G$-fuzzy contraction on $X$ and prove some results concerning the existence and uniqueness of fixed point for such mappings. As a consequence of the main results we derive some extensions of known results from metric into fuzzy metric spaces. Some examples are given which illustrate the results.