Characterization of Gromov hyperbolic short graphs

To decide when a graph is Gromov hyperbolic is, in general, a very hard problem. In this paper, we solve this problem for the set of short graphs (in an informal way, a graph G is r-short if the shortcuts in the cycles of G have length less than r): an r-short graph G is hyperbolic if and only if S _9r (G) is finite, where S _R (G):= sup{L(C): C is an R-isometric cycle in G} and we say that a cycle C is R-isometric if d _C (x, y) ≤ d _G (x, y) + R for every x, y ∈ C.