  Hypergraphs provide a natural way to represent polyadic relationships in
network data. For large hypergraphs, it is often difficult to visually detect
structures within the data. Recently, a scalable polygon-based visualization
approach was developed allowing hypergraphs with thousands of hyperedges to be
simplified and examined at different levels of detail. However, this approach
is not guaranteed to eliminate all of the visual clutter caused by unavoidable
overlaps. Furthermore, meaningful structures can be lost at simplified scales,
making their interpretation unreliable. In this paper, we define hypergraph
structures using the bipartite graph representation, allowing us to decompose
the hypergraph into a union of structures including topological blocks,
bridges, and branches, and to identify exactly where unavoidable overlaps must
occur. We also introduce a set of topology preserving and topology altering
atomic operations, enabling the preservation of important structures while
reducing unavoidable overlaps to improve visual clarity and interpretability in
simplified scales. We demonstrate our approach in several real-world
applications.
