On the Interval of Strong Partial Clones of Boolean Functions Containing

D. Lau raised the problem of determining the cardinality of the set of all partial clones of Boolean functions whose total part is a given Boolean clone. The key step in the solution of this problem, which was obtained recently by the authors, was to show that the sublattice of strong partial clones on {0, 1} that contain all total functions preserving the relation ρ0,2 = {(0, 0), (0, 1), (1, 0)} is of continuum cardinality. In this paper we represent relations derived from ρ0,2 in terms of graphs, and we define a suitable closure operator on graphs such that the lattice of closed sets of graphs is isomorphic to the dual of this uncountable sublattice of strong partial clones. With the help of this duality, we provide a rough description of the structure of this lattice, and we also obtain a new proof for its uncountability.