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A major problem in wireless sensor network localization is erroneous local geometric realizations in some parts of the network due to the sensitivity to certain distance measurement errors, 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 a recent study by the coauthors, an initial formal geometric analysis of flip ambiguity problems has been provided. The ultimate aim of that study was to quantify the likelihood of flip ambiguities in arbitrary sensor neighborhood geometries. In this paper we propose a more general robustness criterion to detect flip ambiguities in arbitrary sensor neighborhood geometries in planar sensor networks. This criterion enhances the recent study by the coauthors by removing the assumptions of accurately knowing some inter- sensor distances. The established robustness criterion is found to be useful in two aspects: (a) Analyzing the effects of flip ambiguity and (b) Enhancing the reliability of the location estimates of the prevailing localization algorithms by incorporating this robustness criterion to eliminate neighborhoods with flip ambiguity from being included in the localization process.