g-approximate best proximity pairs in metric space with a directed graph

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