  We study edge-states in graphene systems where a bulk energy gap is opened by
inversion symmetry breaking. We find that the edge-bands dispersion can be
controlled by potentials applied on the boundary with unit cell length scale.
Under certain boundary potentials, gapless edge-states with valley-dependent
velocity are found, exactly analogous to the spin-dependent gapless chiral
edge-states in quantum spin Hall systems. The connection of the edge-states to
bulk topological properties is revealed.
