  We extend Carter's notion of admissible diagrams and attach a "Dynkin-like"
diagram to each reduced reflection factorization of an element in a finite Weyl
group. We give a complete classification for the diagrams attached to reduced
reflection factorizations. Remarkably, such a diagram turns out to be
cyclically orientable if and only if it is isomorphic to the underlying graph
of a quiver which is mutation-equivalent to a Dynkin quiver. Furthermore we
show that each diagram encodes a natural presentation of the Weyl group as
reflection group. The latter one extends work of Cameron, Seidel and Tsaranov
as well as Barot and Marsh.
