  Breaking inversion symmetry in chiral graphene systems, \textit{e.g.}, by
applying a perpendicular electric field in chirally-stacked rhombohedral
multilayer graphene or by introducing staggered sublattice potentials in
monolayer graphene, opens up a bulk band gap that harbors a quantum valley-Hall
state. When the gap size is allowed to vary and changes sign in space, a
topologically-confined one-dimensional (1D) zero-line mode (ZLM) is formed
along the zero lines of the local gap. Here we show that gapless ZLM with
distinguishable valley degrees of freedom K and K$'$ exist for every
propagation angle except for the armchair direction that exactly superpose the
valleys. We further analyze the role of different geometries of top-bottom
gated device setups that can be realized in experiments, discuss the effects of
their edge misalignment, and analyze three common forms of topological defects
that could influence the 1D ZLM transport properties in actual devices.
