To reinforce the analogy between the mapping class group and the Cremona group of rank $2$ over an algebraic closed field, we look for a graph analoguous to the curve graph and such that the Cremona group acts on it non-trivially. A candidate is a graph introduced by D. Wright... However, we demonstrate that it is not Gromov-hyperbolic. This answers a question of A. Minasyan and D. Osin. Then, we construct two graphs associated to a Vorono\"i tesselation of the Cremona group introduced in a previous work of the autor. We show that one is quasi-isometric to the Wright graph. We prove that the second one is Gromov-hyperbolic. (read more)