Group connectivity of graphs - A nonhomogeneous analogue of nowhere-zero flow properties

Let G = (V, E) be a digraph and f a mapping from E into an Abelian group A. Associated with f is its boundary aS, a mapping from V to A, defined by af(x) = c Dleavingxf(e)-Ceenteringx f(e). We say that G is A-connected if for every b: V-, A with Cx E V b(x) = 0 there is an f: E -+ A (0) with b = af: This concept is closely related to the theory of nowhere-zero flows and is being studied here in light of that theory.