Gromov hyperbolicity of Denjoy domains with hyperbolic and   quasihyperbolic metrics
  We obtain explicit and simple conditions which in many cases allow one decide, whether or not a Denjoy domain endowed with the Poincare or quasihyperbolic metric is Gromov hyperbolic. The criteria are based on the Euclidean size of the complement. As a corollary, the main theorem allows to deduce the non-hyperbolicity of any periodic Denjoy domain. 
Peter H\"ast\"o, Henri Linden, Ana Portilla, Jose M. Rodriguez, E.   Touris