Hyperbolic groups were introduced by Mikhail Gromov in the 80s by considering the geometry of Cayley graphs, viewed as geodesic metric spaces. Several equivalent conditions, due to Gromov and/or Elyahu Rips, are commonly used to characterize hyperbolic geodesic metric spaces. We consider a strengthened version which we call polygon hyperbolicity, and establish equivalent conditions which are variants of the classical alternatives of Gromov. We also characterize those groups whose Cayley graph is a polygon hyperbolic space: they are the finitely generated virtually free groups. These results are joint work with Vítor Araújo (Universidade Federal da Bahia).