The strongly robust simplicial complex of monomial curves

To every simple toric ideal \(I_T\) one can associate the strongly robust simplicial complex \(\Delta _T\), which determines the strongly robust property for all ideals that have \(I_T\) as their bouquet ideal. We show that for the simple toric ideals of monomial curves in \(\mathbb {A}^{s}\), the strongly robust simplicial complex \(\Delta _T\) is either \(\{\emptyset \}\) or contains exactly one 0-dimensional face. In the case of monomial curves in \(\mathbb {A}^{3}\), the strongly robust simplicial complex \(\Delta _T\) contains one 0-dimensional face if and only if the toric ideal \(I_T\) is a complete intersection ideal with exactly two Betti degrees. Finally, we provide a construction to produce infinitely many strongly robust ideals with bouquet ideal the ideal of a monomial curve and show that they are all produced this way.