  We study subcoalgebras of path coalgebras that are spanned by paths (called
path subcoalgebras) and subcoalgebras of incidence coalgebras, and propose a
unifying approach for these classes. We discuss the left quasi-co-Frobenius and
the left co-Frobenius properties for these coalgebras. We classify the left
co-Frobenius path subcoalgebras, showing that they are direct sums of certain
path subcoalgebras arising from the infinite line quiver or from cyclic
quivers. We also discuss the coreflexive property for the considered classes of
coalgebras. Finally, we investigate which of the co-Frobenius path
subcoalgebras can be endowed with Hopf algebra structures, in order to produce
some quantum groups with non-zero integrals, and classify all these structures
over a field with primitive roots of unity of any order. These turn out to be
liftings of quantum lines over certain not necessarily abelian groups.
