Let $(X,d)$ be a metric space endowed with a directed graph $G$ where $V(G)$ and $E(G)$ represent the sets of vertices and edges corresponding to $X$, respectively. We establish sufficient conditions for the existence of a $G$-approximate best proximity pair for a mapping $T$ in the metric space $X$ equipped with the graph $G$ such that the set $V(G)$ of vertices of $G$ coincides with $X$.