We argue the usefulness of Gaifman graphs of first-order relational structures as an exploratory data analysis tool. We illustrate our approach with cases where the modular decompositions of these graphs reveal interesting facts about the data. Then, we introduce generalized notions of Gaifman graphs, enhanced with quantitative information, to which we can apply more general, existing decomposition notions via 2-structures; thus enlarging the analytical capabilities of the scheme. The very essence of Gaifman graphs makes this approach immediately appropriate for the multirelational data framework.