Fig. 3

a A network with \(\mathcal {R}= \{r_t, r_a, r_b, r_1,\dots , r_n\}\). Reaction \(r_i\) with \(i = 2,\dots ,n\) consumes \(p_i\) and produces compound c. \(\mathcal {T}= \{t\}\), \(\mathcal {X}= \{p_1,\dots ,p_n\}\). All stoichiometric values are equal to one. There is one minimal \(SPS\) (\(\{p_1\}\)) and n minimal topological factories in \(\psi (\mathcal {N})\) . One contains only \(\psi (r_1)\). The other minimal topological factories contain each \(\{\psi (r_t), \psi (r_a), \psi (r_b)\}\) and one of the reactions in \(\{\psi (r_2),\dots , \psi (r_n)\}\), respectively. b In this network, the set of compounds is given by \(\mathcal {C}= \{a,b,t,c_1,\dots ,c_n, p_1,\dots ,p_n\}\). The compounds \(p_1,\dots ,p_n\) are the sources and t is the target. The stoichiometric values are equal to 1 if not stated otherwise. Beside the reactions \(r_{a_1}: a \rightarrow t\) and \(r_{a_2}: a \rightarrow b\), there is the reaction \(r'\) that consumes \(n-1\) b and produces \(\{c_1,\dots ,c_n\}\) (1 each). Furthermore, there are n reactions with \({\textit{S}ubs(r_i)} = \{c_i, p_i\}\) and \({\textit{P}rod(r_i)} = \{a\}\), with \(i=1,\dots ,n\). The dots in the Figure illustrate the products \(c_2,\dots ,c_{n-1}\) of \(r'\) that are not shown for simplicity. The reactions \(r_2,\dots ,r_{n-1}\) are not shown for the same reason. There are n minimal topological factories in \(\psi (\mathcal {N})\), each containing the reactions \(\{\psi (r_{a_1}), \psi (r_{a_2}), \psi (r')\}\) and one of the many-to-one reactions of \(\{\psi (r_1),\dots ,\psi (r_n)\}\), respectively. The only minimal \(SPS\) contains all sources