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Reachability preserving compression generates small graphs that preserve the information only relevant to reachability queries, and the compressed graph can answer any reachability query without decompression. Existing reachability preserving compression algorithms either require a long compression time or include redundant data in the compressed graph. In this paper, we propose a novel edge sorting alogrithm for fast, non-redudant reachability preserving compression. On the other hand, we found that the current incremental reachability compression algorithms for dynamic graphs may return incorrect results in some cases. Therefore, we propose two novel incremental reachability compression algorithms. An algorithm called incremental reachability preserving compression with optimum compression ratio, which generates an update compressed graph that is exactly the same as the graph computed by recompression. The other algorithm called fast incremental reachability preserving compression, which can update the compressed graph in a short time. Extensive experiments on real datasets show the efficiency and the effectiveness of our methods. [ABSTRACT FROM AUTHOR]