Constructing reflection-symmetric flexible realisations of graphs

We study reflection-symmetric realisations of symmetric graphs in the plane that allow a continuous symmetry and edge-length preserving deformation. To do so, we identify a necessary combinatorial condition on graphs with reflection-symmetric flexible realisations. This condition is based on a specific type of edge colouring, where edges are assigned one of three colours in a symmetric way. From some of these colourings we also construct concrete reflection-symmetric realisations with their corresponding symmetry preserving motion. We study also a specific class of reflection-symmetric realisations consisting of triangles and parallelograms.