Weakly reversible deficiency zero realizations of reaction networks

We prove that if a given reaction network $\mathcal{N}$ has a weakly reversible deficiency zero realization for all choice of rate constants, then there exists a $\mathit{\text{unique}}$ weakly reversible deficiency zero network ${\mathcal{N}}^{\mathrm{\prime }}$ such that $\mathcal{N}$ is realizable by ${\mathcal{N}}^{\mathrm{\prime }}$. Additionally, we propose an algorithm to find this weakly reversible deficiency zero network ${\mathcal{N}}^{\mathrm{\prime }}$ when it exists.