In the present work, we introduce G-quasi-contractions using directed graphs in metric spaces with a graph and we show that this contraction generalizes a large number of contractions. We then investigate the existence of xed points for G-quasi-contractions under two different conditions and discuss the main theorem. Finally, we list some consequences of our theorem where either the contractive condition is replaced with a stronger one or the underlying space is changed to a complete metric space or a complete cone metric space.
