The neutrosophic graph is a new version of graph theory that has recently been proposed as an extension of fuzzy graph and intuitionistic fuzzy graph that provides more precision compatibility and flexibility than a fuzzy graph and an intuitionistic fuzzy graph in structuring and modelling many real-life problems. The aim of this paper is to offer for the first time the new concepts of neutrosophic highly strong arc, neutrosophic special dominating set, neutrosophic special domination numbers, neutrosophic special cobondage set and neutrosophic special cobondage numbers in the neutrosophic graph, as well as expressing some of relation and results of them and reduce the effect of adding a neutrosophic highly strong arc on neutrosophic special domination number parameter in a neutrosophic graph. Finally, an application related to decision making based on agents affecting the performance of the organization is provided.