  We develop a new framework for investigating linear equivalence of divisors
on graphs using a generalization of Gioan's cycle--cocycle reversal system for
partial orientations. An oriented version of Dhar's burning algorithm is
introduced and employed in the study of acyclicity for partial orientations. We
then show that the Baker--Norine rank of a partially orientable divisor is one
less than the minimum number of directed paths which need to be reversed in the
generalized cycle--cocycle reversal system to produce an acyclic partial
orientation. These results are applied in providing new proofs of the
Riemann--Roch theorem for graphs as well as Luo's topological characterization
of rank-determining sets. We prove that the max-flow min-cut theorem is
equivalent to the Euler characteristic description of orientable divisors and
extend this characterization to the setting of partial orientations.
Furthermore, we demonstrate that $Pic^{g-1}(G)$ is canonically isomorphic as a
$Pic^{0}(G)$-torsor to the equivalence classes of full orientations in the
cycle--cocycle reversal system acted on by directed path reversals. Efficient
algorithms for computing break divisors and constructing partial orientations
are presented.
