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The Domain Name System (DNS) is a critical piece of Internet infrastructure with remarkably complex properties and uses, and accordingly has been extensively studied. In this study we contribute to that body of work by organizing and analyzing records maintained within the DNS as a bipartite graph. We find that relating names and addresses in this way uncovers a surprisingly rich structure. In order to characterize that structure, we introduce a new graph decomposition for DNS name-to-IP mappings, which we term elemental decomposition. In particular, we argue that (approximately) decomposing this graph into bicliques — maximally connected components — exposes this rich structure. We utilize large-scale censuses of the DNS to investigate the characteristics of the resulting decomposition, and illustrate how the exposed structure sheds new light on a number of questions about how the DNS is used in practice and suggests several new directions for future research.