Characterization of graphs equating MC discriminants and PSD algebraic boundaries

Characterize the graphs $G$ for which the matrix completion discriminant $D(G)$ equals the algebraic boundary $partial_a\mathscr C(G)$ of the positive semidefinite completion cone.

Background

For cycle-completable graphs, the paper proves that the matrix completion discriminant coincides with the algebraic boundary of the positive semidefinite completion cone. The equality does not hold for all graphs, as demonstrated by $K_{3,3}$, while it does hold for the non-cycle-completable wheel graph $W_5$.

Because the equality occurs both within and beyond the class of cycle-completable graphs, the paper leaves unresolved the task of determining exactly which graphs have this property.