The characterization of toric vanishing ideals for colored Gaussian graphical models

Let $\mathcal{G}$ be a colored graph, and let $I_{\mathcal{G}}$ denote its vanishing ideal.

Toric vanishing ideal conjecture. $I_{\mathcal{G}}$ is toric if and only if $\mathcal{G}$ is an RCOP block graph.

The preceding theorem proves that RCOP block graphs have toric vanishing ideals. The converse is motivated by examples in which disrupting the block-graph or RCOP conditions produces non-toric vanishing ideals, but it remains conjectural in general.