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In this paper, we study the problem of identifying constant additive link metrics using linearly independent monitoring cycles and paths. A monitoring cycle starts and ends at the same monitoring station, while a monitoring path starts and ends at distinct monitoring stations. We show that three-edge connectivity is a necessary and sufficient condition to identify link metrics using one monitoring station and employing monitoring cycles. We develop a polynomial-time algorithm to compute the set of linearly independent cycles. For networks that are less than three-edge-connected, we show how the minimum number of monitors required and their placement may be computed. For networks with symmetric directed links, we show the relationship between the number of monitors employed, the number of directed links for which metric is known a priori, and the identifiability for the remaining links. To the best of our knowledge, this is the first work that derives the necessary and sufficient conditions on the network topology for identifying additive link metrics and develops a polynomial-time algorithm to compute linearly independent cycles and paths.