Characterize sequentially Cohen–Macaulayness for vertex splittable ideals

Determine whether the sequentially Cohen–Macaulayness of a vertex splittable monomial ideal I ⊂ S = K[x1,...,xn] can be characterized in terms of the ideals I1 and I2 that appear in a vertex splitting I = x_i I1 + I2 of I.

Background

The paper establishes a criterion for Cohen–Macaulayness of a vertex splittable monomial ideal I via a Betti splitting arising from a vertex splitting I = x_i I1 + I2. This yields inductive characterizations for several classes and enables new proofs for known results.

Motivated by this, the authors raise whether an analogous inductive or structural criterion exists for the sequentially Cohen–Macaulay property, formulated directly in terms of the smaller ideals I1 and I2 that arise from a vertex splitting.