Divisors on graphs, orientations, syzygies, and system reliability

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 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 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 systems.