Generalizing Mathlib’s Graph structure to hypergraphs

Determine how to extend the Mathlib Graph structure, which currently uses a vertex carrier set and a binary incidence predicate IsLink: β → α → α → Prop, to support hypergraphs, for example by redefining the incidence predicate as IsLink: β → Set α → Prop.

Background

The paper advocates representing graphs with carrier sets for vertices (and edges for multigraphs) to avoid frequent coercions that arise when modifying graphs (e.g., deletions, contractions, subdivisions). The presented Graph structure uses a vertexSet : Set α and a binary incidence predicate IsLink : β → α → α → Prop, together with symmetry and uniqueness conditions.

To support graphic matroids and broader graph-theoretic developments, the authors consider whether and how this design should be adapted for hypergraphs, where edges can be incident to more than two vertices. They suggest that one natural avenue is to generalize the incidence predicate to relate an edge to a set of vertices but explicitly mark this as unresolved.