The problem of worst case edge deletion from a network is considered. Suppose that you have a communication network and you can delete a single edge. Which edge deletion causes the largest disruption? More formally, given a graph, which edge after deletion disconnects the maximum number of pairs of vertices, where ties for number of pairs disconnected are broken by finding an edge that increases the average shortest path length the maximum amount. This problem is interesting both practically and theoretically. We call it the \emph{single edge deletion problem}. Our contributions include formally defining the single edge deletion problem and providing motivations from network analysis. Also, we give an algorithm that solves the problem much faster than a naive solution. The algorithm incorporates sophisticated and novel techniques, and generalises to the problem of computing the all-pairs shortest paths table after deleting each edge individually. This means the algorithm has deep theoretical interest as well as the potential for even wider applications than those we present here.