We extend the mean field theory-based graphical technique of Cahn [J. Chem. Phys. 66, 3667 (1977)] and include a nearest-neighbor interaction for the surface contribution, in order to identify particular binary mixture-wall combinations that are likely to exhibit critical wetting and duality. Our approach yields a molecular-based understanding of the Schmidt and Binder wetting criteria [J. Phys. (Paris) 46, 1631 (1985)], in terms of the effective wall composition and. We find that systems will exhibit critical wetting only if the wall is energetically equivalent to a "frozen wall," a concept first introduced by Jerry and Nauman [Phys. Lett. A 167, 198 (1992)]. For a second-order transition, the solubility parameter for the wall must lie in between the solubility parameters of the mixture components. Our analysis reveals that systems having a critical wetting transition should also display an enrichment-depletion duality. The verification of duality requires fewer experiments to be performed; thus experimentalists in search of systems exhibiting critical wetting are advised to seek this concomitant phenomenon instead. We show that neither duality nor critical wetting should occur at a free surface (a mixture-vacuum interface).