A graph is said to be absolutely rigid if all its injective realizations in the plane are rigid. A NAC-coloring of a graph is a surjective coloring of edges by red and blue such that every cycle is either monochromatic or it has at least two red and at least blue edges. We will prove that a connected graph has a NAC-coloring if and only if it has a flexible realization (possibly non-injective but with non-zero edge lengths). The set of all NAC-colorings of a graph will be exploited further to state a sufficient condition for the graph being absolutely rigid.