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Erroneous local geometric realizations in some parts of a network due to the sensitivity to certain distance-measurement errors with respect to some neighboring sensor locations is a major problem in wireless sensor-network localization, which may, in turn, affect the reliability of the localization of the whole or a major portion of the sensor network. This phenomenon is well described using the notion of ??flip ambiguity?? in rigid graph theory. In this paper, we present a formal geometric analysis of flip-ambiguity problems in planar sensor networks via quantification of the likelihood of flip ambiguities in arbitrary sensor neighborhood geometries. Based on this analysis, we establish a robustness criterion to detect flip ambiguities in such neighborhood geometries. In addition to the analysis, the established robustness criterion is embedded in localization algorithms to enhance the reliability of the produced location estimates by eliminating neighborhoods with flip ambiguities from being included in the localization process.