*Joint work with Georg Grasegger and Josef Schicho*

We are interested in realizations of graphs in the plane satisfying the constraints given by a labeling of edges by positive real numbers.
A realization is compatible with the labeling if the distance between every two adjacent vertices equals the label.
The labeling is called *flexible* if there are infinitely many compatible realizations modulo translations and rotations.
In the paper Graphs with Flexible Labelings, we have shown that existence of a flexible labeling
is equivalent to existence of a *NAC-coloring* - a coloring of edges by red and blue such that
every cycle is either unicolor, or it contains at least two red and two blue edges.

Nevertheless, even if a NAC-coloring exists, the motion might be degenerated - the compatible realizations are not necessarily injective.
We call a graph *movable*, if there exists a *proper* flexible labeling, i.e., it has infinitely many compatible *injective* realizations.
The concept of *constant distance closure* of a graph is introduced in the paper Graphs with Flexible Labelings allowing Injective Realizations.
It is the graph augmented with extra edges based on NAC-colorings.
We have proved that if a graph is movable, then its constant distance closure is not a complete graph.
Moreover, up to 8 vertices, this necessary condition is also sufficient.
For some graphs, a proper flexible labeling can be constructed based on certain embedding in the space related with two NAC-colorings:
animations of motions with 2 active NAC-colorings