Hibi–Qureshi toric characterization of simple polyominoes

Let $\mathcal{P}$ be a polyomino, and let $I_{\mathcal{P}}$ denote its polyomino ideal. A polyomino $\mathcal{P}$ is simple when it has no holes. The toric ideal of the edge ring of a finite simple graph is the kernel of the monomial map defining that edge ring.

Hibi–Qureshi conjecture. The ideal $I_{\mathcal{P}}$ arises as the toric ideal of the edge ring of a finite simple graph if and only if $\mathcal{P}$ is simple.

Simple polyominoes are known to admit such a toric description through an associated bipartite graph. The conjecture asserts that this phenomenon characterizes simplicity and therefore cannot generally extend to non-simple polyominoes.