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Network reliability analysis is usually based on minimal path or cut enumeration from which the associated reliability expressions are deduced. The cut-set method is a popular approach in the reliability analysis of many systems from simple to complex configurations. The computational requirements necessary to determine the minimal cut-sets of a network depend mainly on the number of minimal paths between the source and the sink. A technique designated as the "Path Tracing Algorithm" is presented in this paper, which can handle both simple and complex networks, and considers both unidirectional and bi-directional branches. A step by step procedure is explained using a bridge-network. The algorithm is easy to program, and does not require limits on the size of the network. The applicability of the proposed technique is illustrated by application to a more complicated system.