We introduce two new methods for approximating the all-terminal reliability of undirected graphs. First, we introduce an edge removal process: remove edges at random, one at a time, until the graph becomes disconnected. We show that the expected number of edges thus removed is equal to (Formula presented.), where (Formula presented.) is the number of edges in the graph, and (Formula presented.) is the average of the all-terminal reliability polynomial. Based on this process, we propose a Monte-Carlo algorithm to quickly estimate the graph reliability (whose exact computation is NP-hard). Moreover, we show that the distribution of the edge removal process can be used to quickly approximate the reliability polynomial. We then propose increasingly accurate asymptotics for graph reliability based solely on degree distributions of the graph. These asymptotics are tested against several real-world networks and are shown to be accurate for sufficiently dense graphs. While the approach starts to fail for “subway-like” networks that contain many paths of vertices of degree two, different asymptotics are derived for such networks.