Dynamic biclique representations for graph updates

Develop an efficient dynamic version of the paper’s biclique-based graph representations that supports both edge insertions and deletions as well as vertex insertions and deletions.

Background

The paper introduces succinct and compact biclique representations that support graph queries such as independent-set, cut, and densest-subgraph operations. These representations are constructed for static graphs, while the densest-subgraph algorithm uses a lazy deletion operation to mark vertices as removed without rebuilding the representation. The authors identify as unresolved whether analogous representations and algorithms can efficiently maintain the data structure under both edge and vertex updates.

References

From the `representation' perspective presented in~\Cref{sec:representations}, an interesting question is whether it is possible to efficiently support dynamic graphs, handling both edge insertions and deletions, as well as vertex insertions and deletions.

— Optimal and Efficient Partite Decompositions of Hypergraphs  (2511.11855 - Krapivin et al., 14 Nov 2025) in Section 6, Further directions, third bullet