The main aim in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph G\. e (respectively, G/. e) obtained from the graph G by deleting (respectively, contracting) an arbitrary edge e from it. A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices. Since we deal with connected graphs, our work provides information about the hyperbolicity constant of minor graphs.