  We study the effect of a structural nanoconstriction on the coherent
transport properties of otherwise ideal zig-zag-edged infinitely long graphene
ribbons. The electronic structure is calculated with the standard one-orbital
tight-binding model and the linear conductance is obtained using the Landauer
formula. We find that, since the zero-bias current is carried in the bulk of
the ribbon, this is very robust with respect to a variety of constriction
geometries and edge defects. In contrast, the curve of zero-bias conductance
versus gate voltage departs from the $(2n+1) e^2/h$ staircase of the ideal case
as soon as a single atom is removed from the sample. We also find that
wedge-shaped constrictions can present non-conducting states fully localized in
the constriction close to the Fermi energy. The interest of these localized
states in regards the formation of quantum dots in graphene is discussed.
