  A toric ideal is called robust if its universal Gr\"obner basis is a minimal
set of generators, and is called generalized robust if its universal Gr\"obner
basis equals its universal Markov basis (the union of all its minimal sets of
binomial generators). Robust and generalized robust toric ideals are both
interesting from both a Commutative Algebra and an Algebraic Statistics
perspective. However, only a few nontrivial examples of such ideals are known.
In this work we study these properties for toric ideals of both graphs and
numerical semigroups. For toric ideals of graphs, we characterize
combinatorially the graphs giving rise to robust and to generalized robust
toric ideals generated by quadratic binomials. As a byproduct, we obtain
families of Koszul rings. For toric ideals of numerical semigroups, we
determine that one of its initial ideals is a complete intersection if and only
if the semigroup belongs to the so-called family of free numerical semigroups.
Hence, we characterize all complete intersection numerical semigroups which are
minimally generated by one of its Gr\"obner basis and, as a consequence, all
the Betti numbers of the toric ideal and its corresponding initial ideal
coincide. Moreover, also for numerical semigroups, we prove that the ideal is
generalized robust if and only if the semigroup has a unique Betti element and
that there are only trivial examples of robust ideals. We finish the paper with
some open questions.
