Least reliable networks and the reliability domination

A well-known model in communication network reliability consists of an undirected graph G whose edges operate independently with the same probability p. Then the reliability, R(G ,p) of G, is the probability that G is connected. It is known that R(G,p) is a polynomial in p and its coefficient of the least-order term is the number of spanning trees t(G), while the coefficient of the highest-order term is the reliability domination d(G) of G. Presented is a complete characterization of graphs that achieve the minimum absolute value |d(G)| over the class of n-node, e-edge connected graphs. Furthermore, the class of graphs that yield minimum t(G) is shown to minimize |d( G)|. The results have applications in the synthesis of least-reliable networks