  To every toric ideal one can associate an oriented matroid structure,
consisting of a graph and another toric ideal, called bouquet ideal. The
connected components of this graph are called bouquets. Bouquets are of three
types; free, mixed and non mixed. We prove that the cardinality of the
following sets - the set of indispensable elements, minimal Markov bases, the
Universal Markov basis and the Universal Gr\"obner basis of a toric ideal -
depends only on the type of the bouquets and the bouquet ideal. These results
enable us to introduce the strongly robustness simplicial complex and show that
it determines the strongly robustness property. For codimension 2 toric ideals,
we study the strongly robustness simplicial complex and prove that robustness
implies strongly robustness.
