This paper focuses on maximum Steiner connected k-core query processing based on graph compression, and proposes a maximum Steiner connected k-core query preserving graph compression algorithm, SC. The correctness of querying based on SC algorithm is proved. Since maximum Steiner connected k-core query only requires a connected component which satisfies certain properties, graph compression algorithm TC is proposed to further compact the compressed graph into a tree. It is proved that querying based on the compacted tree is correct. A novel linear query processing algorithm which is able to query on the compacted tree without decompression is also introduced. Experiments on both real and synthetic datasets demonstrate that the compression algorithm could compress the original graph by 88% in average, and for denser graphs, the compression algorithm achieves better compression ratio, reducing the original graph by nearly 90%. Comparing with the query processing on original graphs, the query performance on compressed graphs is better, and in average, it could be 1 to 2 orders of magnitude times better.