A bypass of an arrow is sectional (Q1203526)

Let \((\Gamma,\tau)\) be a locally finite translation quiver with no multiple arrows or cycles. Given an arrow \(x@>\alpha>> z\), every path \(x = y_ 0 \to y_ 1 \to \dots\to y_ n = z\), with \(n\geq 2\) is called a bypass of \(\alpha\). The main question here is when is it that an arrow admits sectional bypasses. One interesting result implies that, in a connected component \(\mathcal C\) of the Auslander-Reiten quiver of an Artin algebra, if \(\mathcal C\) has no cycles, then every bypass of an arrow is sectional. Another result implies that, if the Artin algebra is of finite representation type, then an irreducible map admits no sectional bypasses.