Techniques herein optimally distribute graph query processing across heterogeneous tiers. In an embodiment, a computer receives a graph query to extract a query result (QR) from a graph in a database operated by a database management system (DBMS). The graph has vertices interconnected by edges. Each vertex has vertex properties, and each edge has edge properties. The computer decomposes the graph query into filter expressions (FE's). Each FE is processed as follows. A filtration tier to execute the FE is selected from: the DBMS which sends at least the QR to a stream, a stream evaluator that processes the stream as it arrives without waiting for the entire QR to arrive and that stores at least the QR into memory, and an in-memory evaluator that identifies the QR in memory. A translation of the FE executes on the filtration tier to obtain vertices and/or edges that satisfy the FE.