Some rainbow problems in graphs have complexity equivalent to satisfiability problems

In a vertex-colored graph, a set of vertices (Formula presented.) is said to be a rainbow set if every color in the graph appears exactly once in (Formula presented.). We investigate the complexities of various problems dealing with domination in vertex-colored graphs (existence of rainbow dominating sets, of rainbow locating-dominating sets, and of rainbow identifying sets), including when we ask for a unique solution: we show equivalence between these complexities and those of the well-studied Boolean satisfiability problems.