Increasing popularity of XML in recent years has generated much interest in query processing over graph-structured data. To support efficient evaluation of path expressions, many structural indexes have been proposed. The most popular ones are the 1-index, based on the notion of graph bisimilarity, and the recently proposed A(k)-index, based on the notion of local similarity to provide a trade-off between index size and query answering power. For these indexes to be practical, we need effective and efficient incremental maintenance algorithms to keep them consistent with the underlying data. However, existing update algorithms for structural indexes essentially provide no guarantees on the quality of the index; the updated index is usually larger size than necessary, degrading the performance for subsequent queries. In this paper, we propose update algorithms for the 1index and the A(k)-index with provable guarantees on the resulting index quality. Our algorithms always maintain a min...