Hyperbolic spaces, defined by Gromov in, play an important role in geometric group theory and in the geometry of negatively curved spaces. The concept of Gromov hyperbolicity grasps the essence of negatively curved spaces like the classical hyperbolic space, Riemannian manifolds of negative sectional curvature bounded away from 0, and of discrete spaces like trees and the Cayley graphs of many finitely generated groups. It is remarkable that a simple concept leads to such a rich general theory. The first works on Gromov hyperbolic spaces deal with finitely generated groups. Initially, Gromov spaces were applied to the study of automatic groups in the science of computation; indeed, hyperbolic groups are strongly geodesically automatic, i.e., there is an automatic structure on the group. The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking. For example, it has been shown empirically in that the internet topology embeds with better accuracy into a hyperbolic space than into an Euclidean space of comparable dimension; furthermore, it is evidenced that many real networks are hyperbolic. A few algorithmic problems in hyperbolic spaces and hyperbolic graphs have been considered in recent papers. Another important application of these spaces is the study of the spread of viruses through the internet. Furthermore, hyperbolic spaces are useful in secure transmission of information on the network. The hyperbolicity has also been used extensively in the context of random graphs. For example, it was shown that several types of small-world networks and networks with given expected degrees are not hyperbolic in some sense. The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; and the references therein. In our study on the hyperbolicity in graphs we use the notations (...)