To prove this theorem, a sufficient condition is first derived in terms of the maximum clique size.
Briefly, the sub-pedigree sets were obtained with the clique-partitioning Jenti method [26], applying different constraints on the splitting procedure (minimum and maximum clique size, minimum relationship level among clique members, and maximum complexity of the resulting families).
As shown in Figure 3, the maximum clique size varies between 2 and 21.
However, as shown in Additional file 2: Supplement SA the number of different subsets, which define the maximum clique size in a given growth condition is considerably less than 2 N.
The maximum clique size of a diversity graph reflects the actual number of the different metabolic patterns regarding by-production and determines the upper bound of the metabolic diversity that emerge in a given growth condition under single-gene knockouts.
Furthermore, the maximum clique size that can be found in a given diversity graph actually determines the upper bound of the metabolic diversity that can emerge in a population in a given growth condition under the specific genetic perturbations.
