On reversal of graph orientation

Let each vertex of a finite directed graph be associated with a finite-dimensional linear space and each arc, with a linear transformation of the corresponding space. Such objects are referred to as linear representations of graphs. They naturally arise in some fields of algebra and are deeply studied in the past three decades.
Replacing all arcs entering into a sink by oppositely oriented ones, we arrive at a new directed graph. These two directed graphs are close to each other in the sense that the problem of classification of their representation are equivalent, as shown by Bernstein, Gelfand, and Ponomarev. Two orientations are equivalent if one is derived from another by means of a sequence of the above transformations.
In the directed graph representation theory, of most interest are circuit-free orientations. In this paper, we give a simple criterion for equivalence of circuit-free orientations. We prove that two orientations are equivalent if and only if some integrals of these orientations are equal to each other.