# Hypergraphic matroid

From Egres Open

Hypergraphic matroids were introduced by Lorea ^{[1]} as a generalization of graphic matroids. Given a hypergraph *H=(V,E)*, a subset *F* of *E* is a **hyperforest** if one can choose two nodes from each hyperedge of *F* so that the resulting graph is a forest. The **circuit matroid** of *H* is the matroid on ground set *E* whose independent sets are the hyperforests. A matroid is **hypergraphic** if it is the circuit matroid of some hypergraph.

If *M* is the circuit matroid of a hypergraph *H=(V,E)*, we call *kM* (the matroid sum of *k* copies of *M*) the **k-circuit matroid** of *H*. The *k*-circuit matroid has rank [math]k(|V|-1)[/math] if and only if *H* is *k*-partition-connected.

## Properties

Although in many respect they are similar to graphic matroids, there are important differences:

- Not all hypergraphic matroids are binary: If [math]|V|=3[/math] and
*E*consists of 4 copies of*V*, we get [math]U_{4,2}[/math] as the circuit matroid. - The class of hypergraphic matroids is not closed under contraction.

## References

- ↑ M. Lorea,
*Hypergraphes et matroides*, Cahiers Centre Etud. Rech. Oper. 17 (1975), 289-291.