  In a Bernal graphene bilayer, carbon atoms belong to two inequivalent
sublattices A and B, with atoms that are coupled to the other layer by
$p_\sigma$ bonds belonging to sublattice A and the other atoms belonging to
sublattice B. We analyze the density of states and the conductivity of Bernal
graphene bilayers when atoms of sublattice A or B only are randomly
functionalized. We find that for a selective functionalization on sublattice B
only, a mobility gap of the order of 0.5 eV is formed close to the Dirac energy
at concentration of adatoms c > 0.01. In addition, at some other energies
conductivity presents anomalous behaviors. We show that these properties are
related to the bipartite structure of the graphene layer.
