*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.

It is well-known, due to Geiringer-Pollaczek and Laman, that a generic labeling a graph is flexible if and only if the graph is not spanned by a *Laman graph*,
where a graph $G=(V_G,E_G)$ is called Laman if

- $|E_G|=2|V_G|-3$, and
- $|E_H|\leq2|V_H|-3$ for every subgraph $H$ of $G$.

We focused on the non-generic cases in the paper Graphs with Flexible Labelings.
Namely, we have shown that existence of a (non-generic) 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 (see animations).

Nevertheless, even if a NAC-coloring exists, the motion might be degenerated - the compatible realizations are not necessarily injective. This happens for instance for this NAC-coloring:

We call a graph *movable*, if there exists a *proper* flexible labeling, i.e., it has infinitely many compatible *injective* realizations.
By “injective”, we mean that no two vertices are mapped to a same point.
A known example of a movable graph is $K_{4,4}$, which has two different motions.

The concept of *constant distance closure* of a graph $G$ is introduced in the paper Graphs with Flexible Labelings allowing Injective Realizations.
It is obtained from the graph $G$ by adding extra edges using NAC-colorings.
We have proven 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.

The proof of sufficiency is based on the known constructions of proper flexible labeling and some new ones: for instance, a proper flexible labeling can be constructed based on certain embedding in $\mathbb{R}^3$ related with two NAC-colorings (see details and animations). For some graphs, we construct a proper flexible labeling from motions of subgraphs or ad hoc.