We report here the reactivity of epitaxial graphene islands and complete monolayers on Ru(0001) towards molecular oxygen and air. The graphene is prepared by thermal decomposition of ethylene molecules pre-adsorbed on an Ru(0001) surface in an ultra-high vacuum chamber. The graphene layer presents a periodically rippled structure that is dictated by the misfit between graphene and Ru(0001) lattice parameters. The periodic ripples produce spatial charge redistribution in the graphene and modifies its electronic structure around the Fermi level. In order to investigate the reactivity of graphene we expose graphene islands to a partial pressure of oxygen and following the evolution of the surface by STM during the exposure. For the exposure to air we removed the sample from the UHV chamber and we re-introduce it after several hours, taking STM images before and after. The surface areas not covered by the graphene islands present a dramatic change but the graphene structure, even the borders of the islands, remain intact. In the case of a complete graphene monolayer the exposure to oxygen or to air does not affect or destroy the rippled structure of the graphene monolayer.