  Combining a variant of the Farkas lemma with the Flow Decomposition Theorem
we show that the regions of any deformation of a graphical arrangement may be
bijectively labeled with a set of weighted digraphs containing directed cycles
of negative weight only. Bounded regions correspond to strongly connected
digraphs. The study of the resulting labelings allows us to add the omitted
details in Stanley's proof on the injectivity of the Stanley-Pak labeling of
the regions of the extended Shi arrangement and to introduce a new labeling of
the regions in the $a$-Catalan arrangement. We also point out that
Athanasiadis-Linusson labelings may be used to directly count regions in a
class of arrangements properly containing the extended Shi arrangement and the
Fuss-Catalan arrangement.
