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Three transformations on networks that reduce the all-terminal network reliability (probability of connectedness) of a network are shown not to increase any coefficient in one form of the reliability polynomial of the network. These transformations yield efficiently computable lower bounds on each coefficient of the reliability polynomial. A further transformation due to Lomonosov is shown not to decrease any coefficient in the reliability polynomial, leading to an efficiently computable upper bound on each coefficient. The resulting bounds on coefficients can, in turn, be used to obtain a substantial improvement on the Ball-Provan strategy for computing lower and upper bounds on the all-terminal reliability. 0 7993 John Wiley & Sons, Inc. NETWORKS AND RELIABILITYA simple model of reliability in communications networks is a multigraph with node set V and edge multiset E; an edge e is an unordered pair of two distinct nodes, and multiple (parallel) edges between the same two nodes are permitted. Loops are omitted without further comment.Each vertex is always present, but edges operate independently, each with the same probability p . The all-terminal reliability Rel(G,p) of such a multigraph G is the probability that G is connected when each of its edges operates independently with probability p.When G contains no multiple edges, it is a simple graph or network.All-terminal reliability by no means captures all of the ingredients of the observed reliability of a network, *On leave from University of Waterloo, Canada.ton leave from University of Saskatchewan, Canada.but has been widely studied as it serves to indicate the suitability of a network topology for supporting network communication.The computation of all-terminal reliability appears to be a computationally difficult problem; Provan and Ball [16] showed that it is #P-complete, and Vertigan [24] showed that it remains #P-complete even for planar networks. For this reason, significant efforts have been made to obtain lower and upper bounds on allterminal reliability that are efficiently computable. We explore this problem further here.Related to the analysis problem of determining (or bounding) all-terminal reliability is the synfhesis problem: Given a number n of nodes and a number m of edges, what is the most (least) reliable connected multigraph (simple graph) with n nodes and m edges? Synthesis and analysis questions have for the most part been treated separately, using different types of techniques. However, it is important to observe that the least (most) reliable network on n nodes and m edges

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