Paradoxical Flexibility of Reflection Symmetric Frameworks

Abstract

A reflection symmetric framework is a graph with an involutive automorphism and a reflection symmetric realization, which is a map from the vertex set to the plane such that the vertices in each orbit are reflection symmetric with respect to a fixed line. Such a framework is called reflection symmetric flexible if it can be continuously deformed while maintaining the symmetry and the distances between adjacent vertices, otherwise it is reflection symmetric rigid. Paradoxical situations may occur in the sense that almost all reflection symmetric realizations of a graph are reflection symmetric rigid, but some special ones are reflection symmetric flexible. The existence of a flexible realization has been characterized combinatorially using the valuations of a function field in the non-symmetric case (Grasegger, Legerský, Schicho, 2019) and rotation symmetric one (Dewar, Grasegger, Legerský, 2022). The reflection symmetric case is significantly more complicated. We provide a necessary condition on the existence of a reflection symmetric flexible realization in terms of the existence of an edge coloring by three colors satisfying conditions on the symmetry and the number of occurrences of the colors in each cycle of the graph. We discuss also some sufficient conditions on the existence of a reflection symmetric flexible realization using these edge colorings. This is joint work with Sean Dewar and Georg Grasegger.