In the present work, we introduceG -quasi-contractionsusing directedgraphs in metric spaces with a graph and we show that this contrac-tion generalizes a large number of contractions.the existence of xed points forWe then investigateG -quasi-contractionsunder two dif-ferent conditions and discuss the main theorem. Finally, we list someconsequences of our theorem where either the contractive condition isreplaced with a stronger one or the underlying space is changed to acomplete metric space or a complete cone metric space.