This paper introduces the Tuple Graph (TuG) synopses, a new class of data summaries that enable accurate selectivity estimates for complex relational queries. The proposed summarization framework adopts a "semi-structured" view of the relational database, modeling a relational data set as a graph of tuples and join queries as graph traversals respectively. The key idea is to approximate the structure of the induced data graph in a concise synopsis, and to estimate the selectivity of a query by performing the corresponding traversal over the summarized graph. We detail the TuG synopsis model that is based on this novel approach, and we describe an efficient and scalable construction algorithm for building accurate TuGs within a specific storage budget. We validate the performance of TuGs with an extensive experimental study on real-life and synthetic data sets. Our results verify the effectiveness of TuGs in generating accurate selectivity estimates for complex join queries, ...