On the Classification of Motions of Paradoxically Movable Graphs
This work addresses a theoretical problem in graph rigidity and motion classification for researchers in discrete geometry and combinatorics, and appears incremental as it builds on known results about NAC-colorings.
The paper tackles the problem of determining all possible proper flexible edge lengths for graphs from their NAC-colorings, using restrictions to 4-cycle subgraphs.
Edge lengths of a graph are called flexible if there exist infinitely many non-congruent realizations of the graph in the plane satisfying these edge lengths. It has been shown recently that a graph has flexible edge lengths if and only if the graph has a special type of edge coloring called NAC-coloring. We address the question how to determine all possible proper flexible edge lengths from the set of all NAC-colorings of a graph. We do so using restrictions to 4-cycle subgraphs.