  Gessel and Sagan investigated the Tutte polynomial, $T(x,y)$ using depth
first search, and applied their techniques to show that the number of acyclic
partial orientations of a graph is $2^gT(3,1/2)$. We provide a short
deletion-contraction proof of this result and demonstrate that dually, the
number of strongly connected partial orientations is $2^{n-1}T(1/2,3)$. We then
prove that the number of partial orientations modulo cycle reversals is
$2^gT(3,1)$ and the number of partial orientations modulo cut reversals is
$2^{n-1}T(1,3)$. To prove these results, we introduce cut and cycle minimal
partial orientations which provide distinguished representatives for partial
orientations modulo cut and cycle reversals. These extend classes of total
orientations introduced by Gioan, and Greene and Zaslavksy, and we highlight a
close connection with graphic and cographic Lawrence ideals. We conclude with
edge chromatic generalizations of the quantities presented, which allow for a
new interpretation of the reliability polynomial for all probabilities, $p$
with $0 < p <1/2$.
