On generalized standard Auslander–Reiten components having only finitely many non-directing modules

Let Γ be a connected component of the Auslander–Reiten quiver Γ(modA) of an Artin algebra. We show that Γ is quasi-directed if and only if the paths in indA, having end-points in Γ, satisfy certain conditions. Moreover, we describe the shapes of such components and identify the classes of algebras to which they are related.