Let $G = (V,E)$ be a simple graph. Then, we can view the set of maximal independent sets (or the set of maximal cliques) as a hypergraph $H = (V, E')$. This is quite a useful device when connecting the (fractional) packing/covering numbers of $H$ with the (fractional) clique/chromatic numbers of $G$ (cf. Scheinerman and Ullman, *Fractional Graph Theory*).

When is the converse true, namely, when does a hypergraph correspond to a set of maximal independent sets?

As an example, $H = ([3], \{ \{1,2\}, \{2,3\} \})$ corresponds to the maximal independent sets of the graph

$1 \text{------}3~~~~~~~~2$,

or equivalently, to the maximal cliques of the complementary graph

$1 \text{------} 2 \text{------} 3$.

But another hypergraph with hyperedges $\{ \{1,2\}, \{2,3\}, \{3,1\} \}$ does not correspond to the maximal independent sets of any graph.

Is there any sufficient and necessary condition that can be captured by the incidence matrix of $H$?