Flexible placements of graphs with rotational symmetry
This addresses a theoretical problem in graph theory and geometric rigidity for researchers in mathematics and computational geometry, representing an incremental advancement by extending known results on flexible placements to symmetric graphs.
The paper tackles the problem of determining when a symmetric graph can be placed in the plane with n-fold rotational symmetry while allowing continuous deformation that preserves symmetry and edge distances, showing that such a flexible placement exists if and only if the graph has a NAC-colouring with an additional symmetry property.
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.