Flexible placements of graphs with rotational symmetry


We study the existence of an $n$-fold rotationally symmetric placement of a symmetric graph in the plane allowing a continuous deformation that preserves the symmetry and the distances between adjacent vertices. We show that such a flexible placement exists if and only if the graph has a NAC-colouring satisfying an additional property on the symmetry; a NAC-colouring is a surjective edge colouring by two colours such that every cycle is either monochromatic, or there are at least two edges of each colour.

In: Holderbaum W., Selig J.M. (eds) 2nd IMA Conference on Mathematics of Robotics. IMA 2020. Springer Proceedings in Advanced Robotics, vol 21.