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Part I derives a new topological formula for the terminalpair reliability of complex networks. The formula generates only non-cancelling terms. The non-cancelling terms in the reliability expression correspond one-to-one with the acyclic subgraphs of the given probabilistic graph. Part II introduces the concept of neutral sequences in acyclic graphs; several of their important properties are established. Based on these results a powerful algorithm for generating the reliability expression is presented. The reliability expression is obtained in symbolic factored form. Examples indicate that the present algorithm is appreciably faster than earlier methods. The properties of cyclic and acyclic graphs established in this paper are significant new results in the theory of digraphs and have further ramifications and wider application than in reliability.