If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle
$T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$
and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic (in the Gromov
sense) if any side of $T$ is contained in a $\delta$-neighborhood of the union
of the two other sides, for every geodesic triangle $T$ in $X$. The study of
hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic
metric space is equivalent to the hyperbolicity of a graph related to it. In
the context of graphs, to remove and to contract an edge of a graph are natural
transformations. The main aim in this work is to obtain quantitative
information about the distortion of the hyperbolicity constant of the graph $G
\setminus e$ (respectively, $\,G/e\,$) obtained from the graph $G$ by deleting
(respectively, contracting) an arbitrary edge $e$ from it. This work provides
information about the hyperbolicity constant of minor graphs.