δ-Hyperbolicity is a graph parameter that shows how close to a tree a graph is metrically. In this work, we propose a method that reduces the size of the graph to only a subset that is responsible for maximizing its δ-hyperbolicity using the local dominance relationship between vertices. Furthermore, we empirically show that the hyperbolicity of a graph can be found in a set of vertices that are in close proximity. That is, the hyperbolicity in graphs is, to some extent, a local property. Moreover, we show that this set is close to the graph's center. Our observations have crucial implications on computing the value of the δ-hyperbolicity of graphs.