We study translations of dyadic first-order sentences into equalities between relational
expressions. The proposed translation techniques (which work also in the converse
direction) exploit a graphical representation of formulae in a hybrid of the two
formalisms. A major enhancement relative to previous work is that we can cope with the
relational complement construct and with the negation connective. Complementation is
handled by adopting a Smullyan-like uniform notation to classify and decompose relational
expressions; negation is treated by means of a generalized graph-representation of
formulae in ℒ+, and through a series of graph-transformation rules which
reflect the meaning of connectives and quantifiers.