Privacy preserving graph publishing is gaining importance in recent times mainly because of inherent privacy issues existing in publishing graph and social network data. Therefore, graph structure needs to be anonymized before publishing. This study proposes anonymization of a graph using fuzzy sets to preserve the graph's privacy while maintaining the utility that can be derived from the graph. We have conducted the experiments on four different datasets, and the results suggest that the proposed approach would not only help in protecting the privacy of data but also in maintaining the quality of data for analysis. To check the robustness of the proposed approach, we have validated the effectiveness of the approach on five key community detection algorithms on three performance measures.