  Using a density functional theory based electronic structure method and
semi-local density approximation, we study the interplay of geometric
confinement, magnetism and external electric fields on the electronic structure
and the resulting band gaps of multilayer graphene ribbons whose edges are
saturated with molecular hydrogen (H$_2$) or hydroxyl (OH) groups. We discuss
the similarities and differences of computed features in comparison with the
atomic hydrogen (or H-) saturated ribbons and flakes. For H$_2$
edge-saturation, we find \emph{shifted} labeling of three armchair ribbon
classes and magnetic to non-magnetic transition in narrow zigzag ribbons whose
critical width changes with the number of layers. Other computed
characteristics, such as the existence of a critical gap and external electric
field behavior, layer dependent electronic structure, stacking-dependent band
gap induction and the length confinement effects remain qualitatively same with
those of H-saturated ribbons.
