  We demonstrate the existence of a new type of zero energy state associated to
vacancies in multilayer graphene that has a finite amplitude over the layer
with a vacancy and adjacent layers, and the peculiarity of being
quasi-localized in the former and totally delocalized in the adjacent ones. In
a bilayer, when a gap is induced in the system by applying a perpendicular
electric field, these states become truly localized with a normalizable
wavefunction. A transition from a localized to an extended state can be tuned
by the external gate for experimentally accessible values of parameters.
