Uniqueness of fixed points for contractions on perturbed metric spaces associated with directed graphs

In this paper, we investigate fixed point results for self-mappings defined on perturbed metric spaces endowed with a graph structure. By combining the notions of a perturbed metric and graph-preserving mappings, we establish a Banach-type fixed point theorem that ensures the existence and uniqueness of fixed points under certain contractive conditions. The presented result generalizes several known fixed point theorems in both standard and generalized metric settings. Illustrative examples are provided to demonstrate the validity and applicability of the main theorem.