  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.
