Graph Theory and Reliability YouMay Rely on the Reliability Polynomial forMuch More Than YouMight Think

The reliability polynomial R p of a collection of subsets of a finite set X has been extensively studied in the context of network theory. There, X is the edge set of a graph VX and the collection of the edge sets of certain subgraphs. For example, we may take to be the collection of edge sets of spanning trees. In that case, R p is the probability that, when each edge is included with the probability p, the resulting subgraph is connected. The second author defined R p in an entirely different way enabling one to glean additional information about the collection from R p. Illustrating the extended information available in the reliability polynomial is the main focus of this article while demonstrating the equivalence of these two definitions is the main theoretical result