The complementation-set correspondence conjecture

Let $G$ be a bicoloured graph with no isolated vertices and no twins, where twins are vertices having identical relevant neighbourhoods. Let $[G]$ denote the parity class of $G$, and let a complementation set be as defined in the paper. Complementation-set correspondence conjecture. The parity class $[G]$ is in bijection with the complementation sets of $G$. The supplied text gives no resolution status and does not provide the definitions needed to specify the bijection more explicitly.