  We study various ideals arising in the theory of system reliability. We use
ideas from the theory of divisors, orientations and matroids on graphs to
describe the minimal polyhedral cellular free resolutions of these ideals. In
each case we give an explicit combinatorial description of the minimal
generating set for each higher syzygy module in terms of the acyclic
orientations of the graph, the $q$-reduced divisors and the bounded regions of
the graphic hyperplane arrangement. The resolutions of all these ideals are
closely related, and their Betti numbers are independent of the characteristic
of the base field. We apply these results to compute the reliability of their
associated coherent systems.
